The proof in BBD is not that complicated, and it doesn't matter much whether $j$ is affine or not. It uses the three following facts :

- If $f$ is a morphism of schemes, then $f_*$ sends a complex of weight $\geq a$ to a complex of weight $\geq a$, and $f_!$ sends a complex of weight $\leq a$ to a complex of weight $\leq a$ (a very natural property of weights; of course that's not so easy to prove for weights of $\ell$-adic complexes, and it is the main result of Deligne's Weil II). Cf BBD 5.1.14.

- If $K$ is an $\ell$-adic complex, then $K$ is of weight $\leq a$ (resp. $\geq a$) if and only if, for every $k\in\mathbb{Z}$, the $k$th perverse cohomology sheaf of $K$ (call it ${}^pH^k K$) is of weight $\leq a+k$ (resp. $\geq a+k$). Cf BBD 5.4.1. Again, hard to prove, but a natural enough property of weights, and a reason in my opinion why perverse sheaves are so much more natural than constructible sheaves (one out of many).

- If $j$ is a locally closed immersion (more generally, a quasi-finite map), then $j_{!*}$ is the image of ${}^pH^0j_!$ in ${}^pH^0j_*$. This is the definition of the intermediate extension.

Now the result you want is obvious : Take $j:X\rightarrow Y$ a quasi-finite morphism. If the perverse sheaf $K$ on $X$ is of weight $\leq a$, then $j_!K$ is of weight $\leq a$ (as a complex), so the perverse sheaf ${}^pH^0j_!K$ is of weight $\leq a$, and so is its quotient $j_{!*}K$. Likewise for weights $\geq a$, using this time $j_*$. 

Note that you could also define $j_{!*}K$ (for $K$ pure of weight $a$) as the weight $\leq a$ part of $j_*K$, or as the weight $\geq a$ part of $j_!K$. I think it's not too hard to recover the usual properties of $j_{!*}K$ from that definition, but I would have to think more to see how to make it work for mixed (but not pure) perverse sheaves.