I don't think there's necessarily something wrong with a felon being an EMT, but would it even be possible for a convicted sex offender? Don't they have to stay away from children? They would have to join a company that only ever did nursing home transports just to stay within the terms of their release.
That depends largely upon state law. In New Jersey, for example, sex offenders, in addition to registration under Megan's Law, are placed on parole supervision for life, and their parole officer would determine whether they could have the job or not.
To reply to another, earlier post, I don't think that a felony record should disqualify you forever from becoming an EMT. Some states, including New York, IIRC, have laws protecting former offenders under these circumstances. This situation actually came up with barbers - you need a state license to be a barber, and the professional boards would deny the license automatically to former felons. Guess what useful skill prisons trained many inmates in? You got it - barbering and cosmetology.
Also, those with only a minor record - say, a single conviction, long ago - may be able to get the conviction expunged or sealed. That is also a matter of state law. Your mileage will vary.
I really has to be a case-by-case analysis by the licensing or certifying authority. Got arrested for lewdness for making a "pressed ham" on a car window? Caught with a joint? Probably shouldn't be disqualifying, regardless of the 'label' (lewdness in some places being a sex offense). Sexually assaulted children? Probably shouldn't be around people, like the sick and injured, unable to defend themselves. Someone with a history of drug crimes applying to be a paramedic, with access to controlled substances? Well, that's going to require a close look and perhaps some monitoring.
But even in those cases, it's possible that there would be variations one way or the other. It really is not a situation where you can draw a bright line rule.
But if you want EMS to be respected as a profession, there should be some sort of backround check involved. Doctors, nurses, school bus drivers, police, firemen, lawyers, accountants, and countless others go through it.