I'll use the $\rho_n$ that are bounded by 1, and $\rho(x,y) = \sum_n \frac{\rho_n(x_n,y_n)}{2^n}$ metric on the product $X = \prod_n X_n$.
Let $O$ be a basic open product set, so $O = \prod_n O_n$, all $O_n$ are open in $X_n$ and where we have a finite set $F \subset \mathbb{N}$ such that $n \notin F$ iff $X_n = O_n$. We want to show it is open in the $\rho$-topology, so pick $x \in O$, and we want to find $r>0$ such that $B_{\rho}(x, r) \subset O$. This would show that all basic product open sets are $\rho$-open, and thus all product open sets are $\rho$-open.
Now, for every $n \in F$, we have that $x_n \in O_n$, which is a (non-trivial) open subset in $X_n$, so we have $r_n > 0$ such that $B_{\rho_n}(x_n, r_n) \subset O_n$, from the fact that the topology on $X_n$ is induced by the metric $\rho_n$. As we have finitely many $r_n$ to consider, we can find $0 < r < 1$ such that $r \le \frac{r_n}{2^n}$ for all $n \in F$.
The claim now is that this $r$ is as required, in the sense that $B_{\rho}(x, r) \subset O$.
To see this, take any $y$ with $\rho(x,y) < r$. For $n \in F$, we know that $\frac{\rho_n(x_n, y_n)}{2^n} \le \rho(x, y) < r \le \frac{r_n}{2^n}$, which implies that for such $n$ we have that $\rho_n(x_n, y_n) < r_n$, and so $y_n \in B_{\rho_n}(x_n, r_n) \subset O_n$. Hence, for all $n \in F$, $y_n \in O_n$, and as the other $O_n$ equal $X_n$ by the form of $O$, we have that indeed $y \in O$, and as $y$ was arbitrary, $B_\rho(x, r) \subset O$, as required.
Now for the other part: we start with an open ball $B_\rho(x,r)$, a basic open subset of the $\rho$-topology, for some arbitrary $x \in X$ and $r>0$, and try to find a basic open subset in the product topology $O$ such that $x \in O \subset B_\rho(x,r)$. This would then show that any $\rho$-open ball is open in the product topology and would show the other inclusion we need: every $\rho$-open set is product open.
The intuition is that the tail of a series like the one that defines $\rho$ is essentially irrelevant (we can get it as small as we like) and this corresponds to the idea that basic open subsets only depend on finitely many non-trivial open sets. So we first pick $N \in \mathbb{N}$ such that $\frac{1}{2^N} < \frac{r}{2}$. This $N$ defines our tail. For $1 \le k \le N$ we consider the open balls $O_k = B_{\rho_k}(x_k, \frac{r}{2N})$, and we set $O_k = X_k$ for $k \ge N+1$.
The claim now is that $O = \prod_k O_k \subset B_\rho(x, r)$, as required. Note that $O$ is indeed a basic open subset in the product topology on $X$ and $x \in O$. To verify the latter claim, we simply estimate: let $y$ be in $O$, then for $k \le N$, $\rho_k(x_k, y_k) < \frac{r}{2N}$, so $$\sum_{k=1}^{N} \frac{\rho_k(x_k,y_k)}{2^k} \le \sum_{k=1}^{N} \rho_k(x_k,y_k) < N\cdot \frac{r}{2N} = \frac{r}{2}\mbox{,}$$ while $$\sum_{k=N+1}^{\infty} \frac{\rho_k(x_k, y_k)}{2^k} \le \sum_{k=N+1}^{\infty} \frac{1}{2^k} = \frac{1}{2^N} < \frac{r}{2}\mbox{.}$$
Putting it together, we indeed get that for $y \in O$ we have $\rho(x,y) < \frac{r}{2} + \frac{r}{2} = r$, as required.
Best Answer
This answer summarizes my comments.
You need to assume $0<\alpha_i\leq 1$ to ensure the new function $\hat{d}$ is a metric, since in that case we have for all $i$: $$ (d_i(x,y)+d_i(y,z))^{\alpha_i} \leq d_i(x,y)^{\alpha_i}+d_i(y,z)^{\alpha_i} \quad \forall x,y\in X_i$$ whereas if we use, for example $\alpha=2$, then we see $$(1+1)^2>1^2+1^2$$
If you have two different metrics $\rho_1$ and $\rho_2$ on a nonempty set $\Omega$ and you show that for every $x \in \Omega$ and every $\epsilon>0$ there are positive constants $\delta_1$ and $\delta_2$ (which may depend on $x$ and $\epsilon$) such that \begin{align} &\rho_1(x,y) < \delta_1 \implies \rho_2(x,y)<\epsilon\\ &\rho_2(x,y)<\delta_2\implies \rho_1(x,y)<\epsilon \end{align}
then the two metrics generate the same open sets, meaning that every union of open balls with respect to one metric is also a union of open balls with respect to the other metric.
For your problem, if you define $\Omega = \times_{i=1}^n X_i$ and you use the fact that $\rho_1(x,y) = \sum_{i=1}^n d_i(x,y)$ is a metric that generates the desired product topology on $\Omega$, you can define $\rho_2(x,y)=\sum_{i=1}^n d_i(x,y)^{\alpha_i}$. Then for each $\epsilon>0$ and each $x=(x_1, ..., x_n) \in \Omega$ you can find $\delta_1>0$ and $\delta_2>0$ that satisfy point 2 above.