When flying VFR without GPS, how do pilots know if they are inside controlled airspace or not?
VFR aviation maps called "sectionals" (and now GPS map displays) depict the types of airspace through borders with different colors and dashed lines. You can buy or download the maps for free from this FAA site.

It is always the responsibility of a pilot to know where they are and follow all applicable laws. In the US, a pilot that breaks a rule because they didn't check NOTAM's can expect certificate action. NOTAM's will inform the pilot of TFR's, "hot" military zones, and other important regional and local (i.e. airport) regulatory information. 

I got my PPL in 1975 and still carry maps with me. I plot an "X" along my route every 10min of flight time (20-30mi). As I reach each checkpoint, I follow a very old axiom, "Never.. Ever.. EVER!.. proceed to the next checkpoint until you identify where you are and what corrections are needed for the next checkpoint". In this manner, you can never be further than 1 or 2min and 1-2 mi from where you should be. 

Even when sightseeing with no particular route, I routinely mark a map about every 10min with my location. In my 2500hrs of flying, the most I have ever been off  in my navigation was 10mi (even when flying 1500mi x-country). So, knowing where you are is not difficult if you practice the skill.

The front of the map has a legend to remind pilots what the coloring stands for.



Sectional map (US Gov public domain)



Front map legend (US Gov public domain)