If you have an Ikea near you, I swear they are the best bag for the money. The bags are under $1 and are very sturdy.
Cons would be:
Top is open (watch for rain, or put a towel on top)
They are almost exactly the size of two square game boxes (Ticket to Ride size) next to each other at the bottom of the bag. They are among the best size and the best value bags that are easy to find.
I get wanting something a bit more heavy duty, but I just wanted to voice this as it works great for me and how often I move games around.
Plastic storage tubs are also popular I have noticed, so that isn't a bad idea.
The Ikea bag has been my bag of choice for well over a year and it has worked out great.