I noticed that a lot of people have this problem, but I probably can't read between the lines and I actually don't know what the problem is about.
I play Starcraft 2 the basic version. In multiplayer matchmaking games I earn no experience at all playing unranked games against people and have not changed from level one since purchasing the game.
Best Answer
What do you mean by the basic version?
If you mean Wings of Liberty, then no you would not be able to earn any experience as the leveling system was introduced in Heart of the Swarm and so you would need this expansion in order to earn experience and level up.
If you mean starter edition, then you should be able to earn experience up to level 5. However since you indicated that you bought the game I doubt that you have the starter edition.
So in order to earn experience you will need to purchase Heart of the Swarm or you will have to be "spawned" which means playing the game in a party with someone who has the expansion.