Hello Colin and welcome to Travelblog!
This question is quite a frequent one with those not only travelling to the USA.
From my prior research, the Visa Waiver Program on the US Government's ESTA website, says the following:
"Have you ever been arrested or convicted for an offense or crime involving moral turpitude or a violation related to a controlled substance; or been arrested or convicted for two or more offenses for which the aggregate sentence to confinement was five years or more; or been a controlled substance trafficker; or are you seeking entry to engage in criminal or immoral activities?"
It then defines "moral turpitude" as such:
"Crimes involving moral turpitude - Such offenses generally involve conduct which is inherently base, vile, or depraved and contrary to the accepted rules of morality and the duties owed to persons or society in general. There are factors, such as the age of the offender or the date of the offense, that may affect whether an offense will be considered a crime involving moral turpitude for purposes of the Immigration and Nationality Act."
Thus, it comes down to two issues, firstly, is non-payment of fines or assault a criminal offence? If not, there should be no problems. Second, even if it is a criminal conviction, then one must look at the other criteria.
I am no expert on this issue, but would doubt a non-payment of fines would cause any issues and suspect that a case of assault 30 years would not either, unless it involved a weapon.
If you want to read more about moral turpitude, please read this from the US Justice Department:
Grounds for Judicial Deportation
There have been long discussions about this previously -
Criminal conviction restrictions? - so by reading this you can see that you are not alone is voicing your concerns or asking your question.
I'm not a lawyer and my experience in observing such matters comes from extensive travel, and I'll try to answer any further questions you have.
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