Did it transform into a name used today or just fall out of use?
Is it connected at all to the name Guy?
In Italian Caio used to be a name until recently, let's say mid-1800 and it's still used when you have to make an example "Tizio says to Caio etc."
Edit: apparently Caius is still used in the US, yet uncommon
In Welsh, or Celtic in general, Caius evolved into Cai or Kei or a similar name, eg, "Sir Kay" in the Arthurian romance. If there was a Kay in the time of the "real" Arthur (if there was a real Arthur), ie, the early 500's, he would have been known as Caius.
Source: mentioned in Marion Zimmer Bradley's "Mists of Avalon", and confirmed on a Name Etymology website.
Cajus Julius Caesar is a member of the Bundestag, the federal parliament of Germany. It is considered a very odd name, but it is still in active use.
At least in Brazil it is widely used, in the form Caio. Personally, I know 3 of them.
In many Latim American countries, 'Caio' is a common name.
If behindthename.com is worthy as a 'source' then the best I can come up with is that Caius/Gaius is related to Kai which in turn is related to Nikolas/Nikolai which is Nicholas in 'English'
Although as mentioned this is using behindthename.com which is debatable as a source
It became "Cayo" in Spanish. The only current public figure that I know of with that name is Cayo Lara, a left-wing politician. It is also commonly used when translating Latin names to Spanish, e.g. "Cayo Julio César".
My former brother-in-law is called Gaius!