Its really a matter of how fast you want to multiplex.
If you are ensuring that only 1 LED at a time will be driven, then you can get by without transistors.
If you want any kind of decent speed tho, you will update a whole layer/plane at a time, and you will need something that can source or sink the current for that whole layer/plane.
If you use a common anode layer, than a PNP transistor or P-channel MOSFET can source the current for the layer, while a power shift register like TPIC6B595 can sink 20mA for each of 8 columns for max brightness.
If you use a common cathode layer, than a NPN transistor or N-channel MOSFET can sick the current for the layer - or maybe even a power shift register, have to look at the specs for other TPIC6x595 shift registers, some will do 350mA/IO pin - but sourcing 20mA/LED is a little tougher, perhaps a High Side switch chip like thishttp://www.digikey.com/product-detail/en/AMIS39101PNPB4RG/766-1004-1-ND/1914838
Many folks use 74HC595 as an 8-output source, but I would speculate that most are over stressed that way.
Adding a PNP array to the output of 74HC595 would be betterhttp://www.digikey.com/product-detail/en/A2982SLWTR-T/620-1127-1-ND/1090370