A straightforward way to do that is a callout, doing a partial mail transaction to see if the putative sender's mail server accepts mail to that address. This approach was popular for a few years, but due to its combination of ineffectiveness and abusiveness, it's now used only by small mail systems whose managers don't know any better.

What's wrong with it?

• The most obvious problem is that it doesn't work to verify addresses. There's lots of reasons why the naively implemented checks that callouts use can succeed for a non-existent address or fail for a good address. Many mail systems (including mine) accept all addresses at the RCPT TO, and reject the bad ones at the DATA command, giving the callout system the mistaken impression that bad addresses are good. Conversely, systems that implement techniques against spam blowback (bounces from spam forging their addresses) can often reject callouts because they look just like blowback, giving the callout system the mistaken impression that good addresses are bad.
• Address verification is not an effective spam filter. These days most spam uses other people's real addresses taken from the spam target lists, so all you're verifying is that they stole someone's address.
• Callouts are abusive. Since upwards of 90% of mail is spam with forged addresses, 90% of your callouts are to people who never sent you mail or otherwise bothered you.
• Callouts are likely to get you blacklisted, since the behavior of callouts is indistinguishable from spammers trying to to do bulk listwashing of bad addresses.
• Callouts are discredited. Every large ISP that used to do callouts, most notably Verizon, found it was counterproductive and stopped.

I was dismayed to hear just last week from an acquaintance who was trying to persuade his system manager not to use this brand new anti-spam silver bullet. I hope he succeeded.