If we have the right definitions, we can have a precise statement which is more or less what you need.
For simplicity let $z = (x,y) \in \mathbb{R}^2$ and denote by
$$X(z) = f(x,y) \frac{\partial}{\partial x} + g(x,y) \frac{\partial}{\partial
y}$$ the vector field corresponding to the smooth (or real analytic
or polynomial) dynamical system
\begin{align*}
\frac{dx}{dt} &= f(x,y)\\
\frac{dy}{dt} &= g(x,y)
\end{align*}
which we will also write as
$$\dot{z} = X(z)$$
For simplicity, we denote by $o = (o_1,o_2) \in \mathbb{R}^2$ the equilibrium
point of $X(z)$, i.e. $X(o) = 0$. Furthermore, by $\phi^t(z)$ we
define the phase flow of $X(z)$. That is $\phi^t(z)$ solves the
initial value problem
$$\frac{d}{dt} \, \phi^t(z) = X\big(\phi^t(z)\big)$$
$$\phi^0(z) = z$$
Definition 1. Let $\gamma = \big\{(x,h(x)) \in \mathbb{R}^2 \,\, | \,\, x \in [a,b]\, \big\}$ be a compact embedded smooth (or real analytic)
curve in $\mathbb{R}^2$. We call it invariant curve of the vector field $X$ whenever
$\bullet$ $\gamma$ is invariant under the flow $\phi^t(z)$, i.e. if $z
\in \gamma$ then $\phi^t(z) \in \gamma$ for all $t\geq 0$, which
is true if and only if the vector field $X(z)$ is tangent to
$\gamma$ for each $z \in \gamma$ and points inwards at the endpoints of $\gamma$.
Definition 2. The invariant curve $\gamma$ of $X$ is called stable whenever
$\bullet$ for any open set $U$ containing $\gamma$, there exists an open
set $V$ containing $\gamma$ such that if $z\in V$ then $\phi^t(z)
\in U$ for all $t\geq 0$. Clearly, $V \subseteq U$.
Definition 3. The invariant curve $\gamma$ of $X$ is called attracting with basin of attraction $BA \subseteq \mathbb{R}^2$ whenever
1. $\gamma$ is stable;
2. $\gamma \subset BA$, where $BA$ is an open subset of
$\mathbb{R}^2$;
3. For any open set $U$ containing $\gamma$ and any $z \in BA$
there exists $T\geq 0$ such that $\phi^t(z) \in U$ for any $t \geq
T$.
4. The open set $BA$ is the maximal open set with properties 1,
2, 3 above.
Similar definitions apply to the point $o = (o_1,o_2) \in
\mathbb{R}^2$.
Definition 4. The equilibrium point $o = (o_1,o_2) \in \mathbb{R}^2$ of
$X$ is called stable whenever
$\bullet$ for any open set $U$ containing $o$, there exists an open set
$V$ containing $o$ such that if $z \in V$ then $\phi^t(z) \in U$
for all $t\geq 0$. Clearly, $V \subseteq U$.
Definition 5. The equilibrium point $o = (o_1,o_2) \in \mathbb{R}^2$ of
$X$ is called attracting with basin of attraction $BA \subseteq
\mathbb{R}^2$ whenever
1. $o$ is stable;
2. $o \in BA$, where $BA$ is an open subset of $\mathbb{R}^2$;
3. For any open set $U$ containing $o$ and any $z \in BA$ there
exists $T\geq 0$ such that $\phi^t(z) \in U$ for any $t \geq T$.
4. The open set $BA$ is the maximal open set with properties $1,
2, 3$ above.
For both definition of attraction, we can show that, due to its
maximality, the basin of attraction $BA$ is an invariant open set for
the vector field $X$, i.e. $\phi^t(z) \in BA$ for any $z \in BA$
and $t \geq 0$.
Theorem. Let $\gamma = \big\{(x,h(x)) \in \mathbb{R}^2 \,\, |
\,\, x \in [a,b]\, \big\}$ be a smooth (or real analytic)
invariant (compact!) curve for the smooth (real analytic or
polynomial) dynamical system in $\mathbb{R}^2$ given by the vector
field $X(z)$ above.
1. Let $o = (o_1,o_2)$ be the only equilibrium point of $X(z)$
lying on $\gamma$;
2. Let $\lim_{t \to \infty} \, \phi^t(z) = o$ for all $z \in
\gamma$.
3. Assume that the invariant curve $\gamma$ is attracting with
basin of attraction $BA \subset \mathbb{R}^2$ (open);
Then the equilibrium point $o$ is attracting
with basin of attraction $BA$ and hence (asymptotically) stable.
Observe that in your example, the set $B$ is not a basin of attraction. It is not maximal. And the equilibrium point is handled only on one side of the invariant curve, while in general you should handle it from all sides (right and left up and down).
Best Answer
The short answer is no -- there is no such general method in dimension >2. Sometimes you can compute one-dimensional separatrices using Melnikov methods, but nothing more. There are also numerical techinques but analysis could be done only for some very specific systems.
Usually you can not determine even one single domain of attraction analytically.