How to donate gold in Diablo 3?

Join their game, run up to them, and right click their portait - there should be a Trade option, which will allow you to trade gold directly to them.

Note that this only works while you're in town.

Trading or gifting gold is no longer possible since February 2014, see: How can gold be traded to other players Diablo 3?

You can still power level them, invite them into your game whenever you find the vault, or gift them items you find while playing in the same party.