I would echo what others have said regarding choosing a pillar drill and generally speaking, a good used one is often far better than a cheap new one.
HOWEVER, I've had an el-cheapo NuTool bench drill for over 30 years which I bought brand new for relative peanuts and which I still use very frequently so it is possible to find a cheap and perfectly usable bench drill and still remain within your budget. I've been threatening to buy a new up-market one for years but resisted the temptation as the NT does everything I need when it comes to a bench drill and owning the latest and best BD wouldn't really improve the work I have for it.
One of the main things I'd look for in any drill press ( new or used) is the stability of the chuck once it is extended to it's maximum drilling depth - with cheap new drills, they often have backlash/side play in the mechanism when fully extended which manifests itself as 'chuck wobble'...new or used, if there is any sign of it then walk away from it as you'll never get accurate/precise drilling and you'll end up buying another once you've become dissatisfied with the results.
A good chuck is essential on new or used drills - on used DP's, very often if worn they can be replaced so they alone are not deal breakers.
Another thing I'd look for is how easy it is to raise and lower the table.... I find that I'm often raising and lowering the table when drilling through timber which is nearly as thick as the maximum travel of the drilling depth. A sticky/stiff mechanism is a pain in the proverbial!
Again don't get too hung up about motor power - it's more about choosing the correct speed ( via the belt/pulley system) rather than raw power alone when it comes to drilling ability. Choosing the correct speed for the material and bit being used, together with a little patience will give far better results than brute force and even modest drill presses will perform quite well if treated sympathetically.
Even with a limited budget, by investigating drills for sale on sites such as Gumtree, it's quite possible to pick up a used bargain which would amply cater for your needs and the beauty is that you can usually take along a few drill bits and scraps of wood to test out your prospective purchase before you commit.