Your question in full: My sister asked me if I could insure her car because her premiums are way higher than mine. In case she has an accident, will my insurance cover her if she is not listed as a driver in my policy? And will my premiums go up on all of my insurance?
First, if your sister is driving any car in your household on a regular basis, then yes, to be properly covered she would normally need to be listed as a driver on your policy. If she has a bad driving record, then this could make your rates rise.
However, what it appears you are thinking about doing is insuring your sister's car (as if it was your own) while she will continue to be the primary driver of the vehicle but not listed as such. This is a very bad idea!
When you insure someone else's car in your own name to obtain lower rates, it is termed "fronting" and normally is illegal.
An insurance company has the right to know who owns the car, the primary driver of the car, and the garaged location of the car. Insurers use this information, as well as other items, to determine if they should take you on as a risk and issue a policy. They also use these as rating factors to calculate your premium amount.
If you are untruthful with your insurer about any information, because the valid information would cause your car insurance company to refuse you a policy or charge you a higher rate, this can be considered material misrepresentation, which is a form of insurance fraud.
By misrepresenting the facts, you may be able to get lower car insurance rates for your sister's car, but the coverages probably won't be there if there is in a car accident. Most car insurance policies include a general provision that excludes coverage for fraud or misrepresentation.
State laws differ for what happens if you are found to you are found to have misrepresented facts. The cancellation of your policy (possibly back to the effective date) is one outcome and denial of claims is another. Depending upon state laws you could also face charges of insurance fraud.
While it may save some money to insure your sister's car, it is unwise to do because the ramifications are severe.
Your sister needs to get her own car insurance policy. Even with a bad driving record, or other issues, your sister should be able to shop around for car insurance and find a policy for her needs. She can start here with a free auto insurance quote online with CarInsurance.com.