Virtually all card issuers support AVS (Address Verification System) in US, UK and Canada. In other countries, it's less reliable. AVS checks only the numeric portion of the street address and postal code, and returns a code indicating whether AVS is supported and whether one or both of the numbers match the ones on file with the card issuer.
Stripe supports AVS the same as other processors. Whether you choose to reject cards that don't pass AVS is up to you: it's a setting. So is which of the codes you want to reject. That's the case at every payment processor I've used. Different businesses, collecting cards in different scenarios, are going to have different risk profiles and different needs. If you're running a store selling computers to strangers over the internet, you probably want full AVS in addition to other fraud checks. If you're taking an invoice payment from another business you've worked with for years, you might not want to bother collecting and verifying an address.
Also, depending on your processor the business' rates may vary depending on how much extra information they provide to help combat fraud. Ex: If you only give credit card info you might be in the highest bracket. If you also give zip code, you'll get dropped down to the next, etc.
Stripe supports AVS the same as other processors. Whether you choose to reject cards that don't pass AVS is up to you: it's a setting. So is which of the codes you want to reject. That's the case at every payment processor I've used. Different businesses, collecting cards in different scenarios, are going to have different risk profiles and different needs. If you're running a store selling computers to strangers over the internet, you probably want full AVS in addition to other fraud checks. If you're taking an invoice payment from another business you've worked with for years, you might not want to bother collecting and verifying an address.