In 2014 at a NY State college I was fired. I was sent a notice in the mail that I was fired due to two separate substantiated claims; one of 'sexual harassment' and the other of 'improper use of school computer.' Now that I am looking for a job I am wondering if this will appear on ANY record a potential employer may search.

Search online, no, unless police/courts were involved.
Call and ask, potentially. Such information is given out at the discretion of the employer.
Most employers have policy regarding what is given out, and you can ask anonymously. Ask in the positive, not the negative.
These days, employers tend to give out less and less because they don't want the liability. Some won't even say if you were fired or quit. They might give dates employed and positions held, and that's it.

Of course none of this applies to gov't jobs requiring security clearance.

If anything does shows up with Sexual Harassment, you will not get the job, particularly at a school..

It is unlikely that the specifics of why you were fired will show up on a background check. But the fact that you worked there, maybe even that you were fired, will show up on a report.

You will need to be prepared to explain these issues.

You don't say what the nature of the harassment was - you can be fired for offensive "harassment" behavior that does not rise to a criminal level. If the police were never notified then there is no official record other than what may be in your personnel file, but in general most employers won't share that information - too big a risk for a lawsuit. However, if the police were notified then it's possible for your new employer to find that out, even if you were never convicted.

You don't say what the nature of the harassment was

The OP stated it was sexual harassment.

The OP stated it was sexual harassment.

True, but sexual harassment in the work place can consist of things like inappropriate jokes, or using offensive images for your screensaver, or inappropriate comments to someone about their body, or pressuring a subordinate for sexual favors. These are all sexual harassment offenses that can get you fired, but are not criminal in nature and would not be reported to the police.