I think Varys makes it perfectly clear that the claimant to the Iron Throne he supports is Aegon. So right off the bat, I have trouble seeing why Varys would have wanted Ser Barristan to go to Dany instead of him. Ilyrio and Varys only support Dany to the extent that she can aid Aegon's claim.
Ser Barristan himself gives Daenarys an account of what happened after he left Joffrey's service. He stayed in King's Landing for a month disguised as a peasant, until he witnessed the execution of Eddard Stark. At that moment, he began questioning his years of service to the Usurper, and realizes the only way to make penitence is to find Aerys' daughter and pledge his service. He never says how he ended up with Ilyrio, but he would have known from the Small Council meetings that Ilyrio and the Targeryen siblings were close.
All indications from Ser Barristan are that he genuinely had a desire to seek Dany out and protect her. He could be lying I suppose, but we get no sense of that from his POV chapters and it'd run contrary to his character. So if Varys DID intend for Ser Barristan to go to her, it was a pretty lucky guess on his part.
Meanwhile, I can see many reasons why Varys wouldn't want Barristan with Dany. Ser Barristan has a very negative view of Varys, and outs Ser Jorah as one of the Spider's spies. He can also confirm that Varys was the one who sent Robert's order to have her killed, and even voted for her death.
I think it's much more likely that Varys just wanted to weaken Joffrey by sending away a major asset. The fact that Ser. Barristan ended up with Daenarys instead of Renly or Stannis was likely just a happy coincidence.