Can I legally give my American boyfriend money?

My boyfriend has a debt that I want to help him pay off. Legally can I take out cash or pay for this debt without running into issues? (ie: leaving money in the USA). I'm here as a short term visitor.

Not a problem. If it is over 100K, he has to report it on a form to the IRS (because you are a non-resident), but no tax is owing, and you don't have anything to report in any event.

Over 10K needs to be reported at the Border or by Mail.

Just to clarify, the 10K or more would only need to be reported to customs if you actually carry the cash across the border. for transfers done through the banking system, the banks take care of any reporting requirements.