I don't think 302's will hurt your rankings. But you could potentially be (and likely are) wasting link juice which would help your rankings.
Think of it like this...are you hurting your presentation by not using eye catching, memorable graphs and charts? No...But you could potentially make your presentation better by adding these things.
It's about potential, not right/wrong...You're not doing something wrong by using 302's, i.e. you won't get penalized for it. But, all things being equal, 9/10 your competitor whose using 301's will outrank you.
My general rule of thumb is if I can be doing something that could likely help me, then I do it. It's much easier on the mind.
The way you fix them depends on how they were setup. Were they setup through manual htaccess modifications, a plugin, a tool in CPanel or some other hosting platform, etc?
I would suggest changing them them the same way they were created to prevent confusion, redirect loops, etc.
