Scroll down the page to watch some great guitar players who use their thumb to play the bass notes of chords...and skip my intro rant 🙂
Playing Chords With Your Thumb is Wrong...Not
It's kind of funny, but there is a segment of guitar players and guitar teachers who have this misguided idea that you should NEVER do this. Play chords where you play notes with your thumb. The evil "thumb chords".
D/F# Chord - Sorry, This Evil "Thumb Chord" is Used All The Time
This chord is used in "Freebird" by Lynyrd Skynyrd. Are you really going to argue with Freebird?
Left Hand Classical Guitar Technique is the Only Valid Way to Play Guitar...Not
I get it. I studied classical guitar for years. My first guitar lessons at age 9 were with a college classical guitar teacher.
I understand about the times when there is an advantage to that hand position. The left leg on a footstool. Guitar at the perfect angle. Thumb in the middle of the back of the neck and all 4 fingers having the best reach possible.
Watch Master Classical Guitarist Andrés Segovia Play

But sometimes you ain't playin' Bach. OK, well technically he was playing Scarlatti and Sor in that video, but I'm makin' a point here. For some things this narrow idea of what is the only proper way to hold and play the guitar just does not makes sense.
One of those things is playing chords where you play a note on the 6th string with your thumb. There are chord sounds that you just cannot get any other way.
See the John Mayer Video Lower...Very Cool Thumb Chords
It depends on what style of music you are playing. It can also depend on the guitar you are playing. Classical guitars have wide necks. So even if you wanted to play a note on the 6th string with your thumb, it would not be easy. But that guitar is built for playing classical music.
Raise a stink about it if you must. Hey everyone has their right to an opinion. But I will let some of the guitar players who have made good use of using their thumb while playing chords speak for themselves.
John Mayer
John makes uses of all kinds of chords where he plays notes on the 6th string with his thumb. Many could not be played any other way and helps give his acoustic songs a unique sound.
Below is acoustic guitar show that he played back in 2008. Keep an eye on this left hand and all of those notes he is playing with his thumb.
John Mayer - 2/4/08 - Private Acoustic Show in The Bahamas w/ Robbie McIntosh

Richie Havens
We can't talk about playing the guitar with your thumb without talking about Richie Havens. Richie had a completely unique way to play the guitar. He used an open-D tuning in conjunction with chord forms where he play all kinds of notes with his thumb. Not just the 6th string.
A lot of people know Richie Havens from his song "Freedom" that he played at Woodstock. But below is a concert he played in 1974 where you can see more of his unique way of playing guitar.
Richie Havens in Concert 1974

Glen Campbell
Glen certainly made use of his thumb on some chords. You can catch him doing so in this classic performance of "Wichita Lineman". Watch the first chord he plays. Busted, thumb chord.
Glen Campbell Wichita Lineman Live in 1985

Jimi Hendrix
I so often want to go back in time beat the cameramen who zoomed up on the faces of great guitar players and rarely showed their hands. This is definitely the case with Jimi Hendrix. Often the footage of him does not show his hands very often. But you can see him using some chords with his thumb in this early video of him playing "Hey Joe".
Jimi Hendrix at the Marquee

Yes, you should be careful if you are going and never play chords that are hurting your hands. Know the difference between pain and discomfort. Thumb chords can take some work to get used to, but you can open up a whole new world of sound in your chords by doing so.
If you want to expand your chord vocabulary, including thumb chords...check out my comprehensive Rhythm Guitar Mastery course.