Shulchan Aruch Even Haezer 21- 5
It is forbidden to be served by a woman at all, whether she is an adult or a minor, whether a slave or free, lest he come to have forbidden thoughts. What service were they speaking of? Washing his face, hands, and feet,--
Rema. Some say that all this is forbidden only when they are alone, but in a place like a bathhouse where many people are found, one may be washed by an idolatrous slave woman, and this is the custom. And some say that anything not done in an affectionate manner where his intent is only for Heaven's sake, is permitted. Thus the custom is to be lenient in these things.
Bach Quotes the Maharshal “ For this reason the minhag is to dance with the Kallah (hand in hand)
so that her husband will love her or for the kovod of her father, however, Talmidei Chachamim,
should refrain” (dancing with the kallah)
The Bach adds that in his country (Galicia) “the minhag was, even for Gedolim to dance with the
Kallah. Where the minhag is to dance you keep the minhag, where there is no such minhag
then there is an issur (to dance with the Kallah).”
The Poskim who strongly opposed this minhag, (hand in hand) call this a mitzvah haboh beaveiro.
Eventually the Rabonim managed to abolish this minhag of dancing hand in hand with the kallah
and instituted instead to dance with a napkin held by the kallah.
For the father and grandfathers of the kallah there never was a problem holding hands and
therefore the minhag continues on until today.
The Chasidim replaced the napkin with a gartel, probably to keep a further distance from the
Kallah or could be they had no napkins.
Dancing and being measmeach the kallah is a Mitzvah (Tur –mitzvah gedolah) and by using
a gartel or napkin it becomes a MITZVAH instead of a Mitzvah Haboh Beaveiro.
For those who who never had the original minhag (hand in hand) there was no need to
institute the new minhag of using a napkin or gartel. Hence the opposition of the
Yeshivishe people for todays Mitzvah Tanz.