There are no specific requirements at a state or federal level that dictate you need to insure your aircraft. That being said, many FBOs (fixed base operators) will ask for proof that your aircraft is insured before you can use their facilities.
There are many types of aircraft insurance that cover various potential issues. You can get in-flight insurance, public liability insurance, passenger liability insurance, combined single limit insurance, aircraft hull insurance, and crew coverage.
Many insurance companies will require the pilot to have some kind of FAA medical certification. Some companies specify that this can be a Basic Med qualification but this varies between insurance brokers.
We would always recommend reaching out to your specific insurance provider for further details. They will be able to advise you further on whether Basic Med certifications fit with your insurance policy.
Generally speaking, Basic Med will be accepted by most insurance providers. The caveat to this is that if you are an older pilot (70 or over) you are likely to still need to provide your insurance broker with an FAA medical certificate. Another instance where you may need more than Basic Med is if you are flying a high performance aircraft.