A dissenting opinion:

We used to install a fair amount of that style basket during the summer (beach town). I agree with Andy that a quick release is a worst-case scenario with that mounting option, but it can be mounted quite securely using either front dropout braze-ons (if present), and sometimes rear-facing braze-ons (fender mounts), if the stays can clear the front hub locknuts. If neither of those methods work, a large-ish hose clamp and nut/bolt is still fairly secure; I would load a 5 pound weight in the basket while setting up to make sure that it would sit in the same spot on both sides. 7-8mm wide washers help provide a good amount of clamping force.

All that said, I agree with Andy, but for a different reason: you have a suspension fork, however basic, on that bike, so the lower struts will not have a secure place to attach, no matter what. If you really want a basket on that bike, I would recommend you look at the Wald 3133. While it has a slightly lower weight limit, it relies solely on the stem/handlebars for support.