San Francisco has Golden Gate Park, which is bigger and prettier than Central Park. Plus, San Francisco has several other parks that are even nicer. Both cities have better parks than those... but again, SF has more and they're overall much nicer. If your decision is based on parks, SF is definitely the better choice (just look at a map).
New York, per population, is actually safer than San Francisco. 5 years ago this wasn't true... but the crime in New York has recently gone way down, vs SF which has gone way up. To compare, NY has about 400 homicides per year (a fraction of what it used to be... used to be really dangerous) relative to a population of about 8 million. SF has a bit over 100 per year (about 3 times the amount it was, say, 10 years ago) in a population of under 800,000. Neither city is terribly dangerous as far as big cities go, but New York is definitely safer overall. EDIT: Um, seems SF's murder rate actually drastically decreased in 2009! New York's did too, though. Haven't done the math so not sure which is actually statistically safer, but *assuming this trend continues*, both are apparently now super safe for big cities! Could be SF's increase was just a 3 or 4 year anomaly.
If you just want a simple, cheap place to live, neither city is a good option. However, New York (sans Manhattan) is more conducive to finding a cheap place than San Francisco. Finding a job in any major US city is tough right now, so hopefully you're not graduating for a while.
As far as where to live in the general sense... it's a matter of taste. There's no shortage of SF/NY comparisons online, so read up. However, if you want the full urban/big city/America in all its opulent glory experience, go for New York. The stereotype is huge, busy, loud, rude, fast, yet awe-inspiring. It's grandiose and what American movies/big city stereotypes are made of. San Francisco is relatively quiet and slow compared to New York, is much prettier and more romantic/poetic in appearance and feel than NY (relatively small Edwardian/Victorian/Deco city, built on hills, surrounded by water, the headlands directly north... quite pretty), nicer weather, etc... However, it IS very smug, PC (and self-fellating in its PC-ness, at that), and kinda full of itself. Less direct than NY, for better or worse (depends on preference, I guess). But that's not saying anything you can't find a million other places... so read up! New York is overall less expensive (to rent a place, anyway) but also has far more crappy areas (hence the cheaper price).