There are tons of options. Do a KLH model Six search here on AK and threads will come up that will likely include some examples of pairings that others have as examples of options. The turntable does not matter as much with the speakers as the amp/receiver would. Also, if you dont need AM/FM radio, consider an integrated amp.
I have personally connected KLH six's, 23's, 17's etc up to several different pieces and they all sounded pretty good. I had a nice Pioneer SX737-KLH model 17 combo that sounded wonderful. Many of the mid power receivers or integrated amps in the various lines by Pioneer, Marantz, Sansui, Kenwood, Yamaha etc all would be nice. Just find one known to have a good phono section. I have a Pioneer SA-9100 Integrated that is fantastic in many regards and super versatile.
Good recommendation above too for tube gear which always sounds good with the likes of the KLH Six. Will just be a bit more difficult to come across and manage. But a lot of fun if you are willing to get into it.
In all of these think carefully before purchasing anything vintage. it may come with issues or some might arise. It may also not sound as good as it should. Look for restored pieces. They may be more $ up front, but may be worth it. it is the same with your Six's. They may benefit from some refurbishment.
Post a few photos of them if you can including the rear. Do the grilles come off easily?