Do the Sikhs believe that Krishna existed?
Not in the same way as most Hindus.
Sikhs do not believe in avatars as God coming down in a human form, so Krishna is usually nothing more than a wise or saint man to Sikhs.
Sikhism believes there have been many:
ਕੇਤੇ ਪਵਣ ਪਾਣੀ ਵੈਸੰਤਰ ਕੇਤੇ ਕਾਨ ਮਹੇਸ ॥
So many winds, waters and fires; so many Krishnas and Shivas.
- Page 7; Gurū Nānak Dev
ਹਰਿ ਆਪੇ ਕਾਨ੍ਹ੍ਹੁ ਉਪਾਇਦਾ ਮੇਰੇ ਗੋਵਿਦਾ ਹਰਿ ਆਪੇ ਗੋਪੀ ਖੋਜੀ ਜੀਉ ॥
The Lord Himself created Krishna, O my Lord of the Universe; the Lord Himself is the milkmaids who seek Him.
- Page 174; Gurū Rām Dās
And also uses Krishna etc. as metaphors:
ਆਪੇ ਗੋਪੀ ਕਾਨੁ ਹੈ ਪਿਆਰਾ ਬਨਿ ਆਪੇ ਗਊ ਚਰਾਹਾ ॥
The Beloved Himself is the milk-maid and Krishna; He Himself herds the cows in the woods.
- Page 606; Gurū Rām Dās
Or as mere names of the Lord:
ਏਕ ਕ੍ਰਿਸਨੰ ਸਰਬ ਦੇਵਾ ਦੇਵ ਦੇਵਾ ਤ ਆਤਮਾ ॥
The One Lord Krishna is the Divine Lord of all; He is the Divinity of the individual soul.
- Page 469; Gurū Angad Dev
Hope this clears it up a bit.