This is a good question, but at first blush it is hard to answer. This is because there is always an ambiguity in where you put the meaningful definitions - you can put it in the force constant, or you can put it in the units for the magnetic charges.
For magnetic monopoles, the first place one naturally turns to is to the magnetic Gauss law, which would be modified to the form $\nabla\cdot\mathbf B\propto \rho_\mathrm{m}$ ... except that you don't really have a way to fix that proportionality constant. By symmetry with respect to the electric Gauss law ($\nabla\cdot\mathbf E=\rho_\mathrm e/\varepsilon_0$) you'd hope that would be $\mu_0$, but that's some hardy guessing.
What you do want, very much, is for magnetic monopoles to have the same relationship to magnetic dipoles as electric monopoles have with electric dipoles - and magnetic dipoles we know how to handle.
If you take this as your basis, then, you can postulate* a magnetic field of the form
$$
\mathbf B=\frac{k_1q_\mathrm{m}}{r^2}\hat{\mathbf r}, \tag 1
$$
with some as-yet-to-be-determined constant $k_1$, for a magnetic monopole of magnetic charge $q_\mathrm{m}$. The key physical input is requiring a pair of opposite magnetic charges a distance $d$ apart to have a magnetic dipole moment of
$$
m=q_\mathrm{m}d,
$$
as in the dipole case, and this fixes the units of the magnetic charge: you know that this magnetic dipole (made from two opposite magnetic monopoles) must be exactly equivalent to a standard magnetic dipole: a current $I$ in a loop of area $A$, with magnetic dipole moment $m=IA$. This requires you to have $[q_\mathrm{m}]=\mathrm{A\,m}$.
From here you can get the dimensionality of the magnetic field constant:
$$
[k_1]
=[Br^2/q_\mathrm{m}]
=[BL/I]
=\left[\frac{\mu_0I}{L}\frac{L}{I}\right]
=[\mu_0]
=\mathrm{N/A^2}.
$$
However, it doesn't tell you the numeric value of this constant, for which you need to go back to your initial physical input - the equivalence of opposite pairs of magnetic monopoles with current loops. In particular, if each magnetic monopole has a magnetic field as in $(1)$, then it follows that a pair of them, with opposite magnetic charges and separated by a distance $d$ along an axis $\hat{\mathbf u}$, must have the magnetic field
$$
\mathbf B=\frac{k_1q_\mathrm{m}d}{r^3}\left(3(\hat{\mathbf u}\cdot\hat{\mathbf r})\hat{\mathbf r}-\hat{\mathbf u}\right).
$$
This contrasts with the magnetic field of a current-loop magnetic dipole with magnetic dipole moment $\mathbf m$
$$
\mathbf B=\frac{\mu_0/4\pi}{r^3}\left(3({\mathbf m}\cdot\hat{\mathbf r})\hat{\mathbf r}-{\mathbf m}\right),
$$
which needs to be equivalent under the identification $\mathbf m=q_\mathrm{m} d\:\hat{\mathbf u}$, and this completely fixes the magnetic-field constant at
$$
k_1=\frac{\mu_0}{4\pi},
$$
i.e.
$$
\mathbf B=\frac{\mu_0}{4\pi}\frac{q_\mathrm{m}}{r^2}\hat{\mathbf r}
$$
for a point magnetic monopole of magnetic charge $q_\mathrm{m}$. Similarly, this fixes the magnetic Gauss law to the naive
$$
\nabla\cdot\mathbf B=\mu_0\rho_\mathrm m
$$
(where $\rho_\mathrm m$ is of course the volumetric density of magnetic charge).
OK, so that's a lot of work - and we're nowhere near the force that you asked about. The reason for this is that we can only get out of the formalism what we put in, and we have only specified how magnetic monopoles should produce magnetic fields, but not how they should react to them. To get that from the formalism, we need to give it more information, and here again we are constrained in that a opposed-point-monopoles magnetic dipole needs to feel exactly the same force that a current-loop magnetic dipole of the same magnetic moment does.
Similarly to the above, we can postulate* that an external magnetic field $\mathbf B$ will produce a force
$$
\mathbf F=k_2q_\mathrm m\mathbf B
$$
on a point magnetic monopole of magnetic charge $q_\mathrm m$, and see what happens. Given this postulate, the same algebra that worked for electric dipoles implies that if you have two magnetic monopoles of opposite magnetic charges $q_\mathrm m$ a distance $d$ apart along the unit vector $\hat{\mathbf u}$, then the torque on them exerted by an external uniform magnetic field $\mathbf B$ will be
$$
\boldsymbol{\tau} = (k_2q_\mathrm md\:\hat{\mathbf u})\times\mathbf B,
$$
which contrasts with $\boldsymbol{\tau} = \mathbf m\times\mathbf B$ for a usual current-loop magnetic dipole of magnetic dipole moment $\mathbf m$. This equivalence then forces our second constant to be
$$
k_2=1.
$$
Similarly, you can check that this force will give identical expressions for the force on a magnetic dipole in a non-uniform external magnetic field $\mathbf B(\mathbf r)$, regardless of whether it is made up from opposing magnetic monopoles or from a small current loop.
With this, then, we can just connect the two main laws - how magnetic monopoles produce magnetic fields and how they react to them - to get the answer we're after. If you have two magnetic monopoles of magnetic charges $q_{\mathrm m,1}$ and $q_{\mathrm m,2}$, separated by a distance $r$ along the separation vector $\hat{\mathbf{r}}_{1\to2}$ pointing from $1$ to $2$, then the magnetic force exerted by magnetic monopole $1$ on magnetic monopole $2$ is
$$
\mathbf F=\frac{\mu_0}{4\pi} \frac{q_{\mathrm{m},1}q_{\mathrm{m},2}}{r^2}\hat{\mathbf{r}}_{1\to2}.
$$
As expected, like poles repel each other (e.g. two point north poles repel).
Here the magnetic charges are measured in ampere meters, and the magnetic charges can be calibrated by observing the interaction with a moving electric charge: if you have a point magnetic monopole of magnetic charge $q_\mathrm{m}$ and an electric charge $q_\mathrm{e}$ separated by a distance $r$ along the unit separation vector $\hat{\mathbf{r}}_\mathrm{m\to e}$, with the electric charge moving at velocity $\mathbf{v}_\mathrm{e}$, then the electric charge will be subject to a force
$$
\mathbf F=\frac{\mu_0}{4\pi} \frac{q_{\mathrm{m}}q_\mathrm{e}}{r^2} \mathbf{v}_\mathrm{e} \times \hat{\mathbf{r}}_\mathrm{m\to e}.
$$
This gives you an 'anchor' for the unit of magnetic charge - it's not free-floating, and it's completely tied to the unit of electric charge.
* Note that these are also additional impositions on the form of the produced magnetic field and the response to external fields. However, both postulates are reasonable things to assume: if the relations were substantially different, then we wouldn't want to call those objects magnetic monopoles. Moreover, it should be essentially possible to get to those forms using very little more than symmetry considerations.
Since the magnetic monopole acquires a significant amount of energy, you cannot just treat it like a (magnetic) test charge that moves passively on and on in the supposedly constant solenoidal magnetic field. Only the whole system encompassing the solenoidal magnetic field, the electric current that creates it, the moving monopole and finally the solenoidal electric field that will be generated by it, must obey energy conservation.
The situation is easier to understand, if you take an infinite straight line current as the generator of the original magnetic field. Then the magnetic flux density at a distance $r$ from the line current has constant magnitude and its field lines are strictly circular.
Now place a cylindrical shell of magnetic monopoles around the electric line current at radius $r$. Then these monopoles will start moving circularly around the line. By symmetry of Maxwell's equations with magnetic monopoles, these circularly moving monopoles are similar to an infinitely long "coil" in electrical engineering, and hence, will generate a homogeneous electric field inside of them, which is then, of course, directed in parallel to the electric line current that generated the original magnetic field. Hence, this additional electric field will decelerate the line current and weaken the magnetic field, which in turn leads to attenuated acceleration of the magnetic monopole shell.
After all, potential energy of the original magnetic field and kinetic energy of the initially moving charges will be converted into potential energy of a newly generated electric field and kinetic energy of the moving magnetic monopoles. Total energy will stay the same, however.
When you take this simple example to your non-ideal magnetic dipole field and a magnetic point charge instead of a shell, only the details get more complicated, while the principles stay the same.
Basically, this is all pretty similar to Lenz' rule in classical electrodynamics, only adding magnetic monopoles: the effect of induction (when applied to magnetic monopoles) counteracts its cause and so makes sure that energy is conserved.
Additional note: be aware of the fact that classical electrodynamics (i.e. without magnetic monopoles) also allows closed electric field lines. Although these only occur during radiation, they principally exist for a certain duration. If your argument were true, you could also apply it to closed electric field lines, and so energy conservation would be violated already for Maxwell's theory. But of course, energy is also conserved there, again because of the "test charge" acting back on the "external field".
Yet another note: Actually, energy conservation is a surprisingly non-trivial topic in classical electrodynamics. While in classical mechanics, you are always assuming a complete Lagrangian from whose explicit time-independence you can derive energy conservation, there is no such complete Lagrangian for classical electrodynamics with massive charges. This is known as the problem of radiation damping, and it is one of the things that motivated the development of quantum mechanics. In quantum electrodynamics radiation damping can be treated consistently. But until the discovery of quantum electrodynamics, physicists had to assume that energy was "just conserved" although they could not strictly prove it from theory (at least not in the presence of charged masses). On the other hand, if there are no charges/masses, energy conservation in classical electrodynamics is trivial, but then again, what use is a "dry" theory of vacuum fields that cannot act on something material?
Best Answer
Blas Cabrera designed and built a magnetic monopole detector. Here's one report on it:
(Old link at http://www.slac.stanford.edu/cgi-wrap/getdoc/ssi82-025.pdf now dead and not archived by the Wayback Machine.)