Before midnight:
I'm assuming you already have an Istanbul card with credit in it.
Also see: How does public transportation work in Istanbul?
First, check the ferry schedule from Karakoy (be careful it changes seasonally). Go to Tunel (underground funicular) at the bottom of Istiklal Avenue. Underground funicular is where the Tunel Square is. Istiklal Avenue meets a square before you go into a tiny street that has a lot of stores with musical instruments.
Take the Tunel to Karakoy. You may wait a little for the Tunel but it will take around 2 minutes to get to Karakoy (around 1.75TL).
Then, walk to Karakoy Ferry Station. You need to cross the street (don't cross the Galata Bridge), and you will see it there. Check out the map.
The ferry will cost around 1.75TL, and will take 20min to get to Kadikoy Iskele (pier).
After midnight:
I'm assuming you have cash.
You have one option: Dolmus (yellow minibus for 8 people).
They are parked next to AKM (Ataturk Cultural Center-picture). It costs around 6-7 TL. Make sure you ask the people there because there are 3 types of Dolmuses that go to Asian side. a) to Kadikoy, b) to Bostanci (through the Caddebostan-shoreline on the Asian side), to Bostanci (through the minibus avenue on the Asian Side)
This option still exist before midnight, however it wouldn't be the fastest since the Bosphorus traffic is pretty miserable during rush hour. If you are in Taksim Square, and you think that the traffic will be light at the time you are commuting, then go with the Dolmus.
