Sending data through lasers
Now what I was thinking was to check if the laser light is on then '1' is passed and when the laser is off '0' would be passed.


This is exactly how most fiber optic communications works.


  How would I check for [if a 1 or a 0 was passed?]


You would build a photoreceiver circuit.

This typically involves a photodetector, possibly an automatic gain control amplifier, and a decision threshold amplifier. 


  another problem I'm facing is if it's on it may keep on passing 1's


In fiber optics data communications, this is typically solved by using an encoding scheme such as 8b/10b which ensures a minimum transition density. The trade-off is of course that you need to send 10 symbols for every 8 bits in your actual message. Higher-speed systems use more complex encoding schemes with lower overhead.

In telecommunications systems, scrambling is used so that, although there is no guarantee, the odds of a long run of 1's or 0's being sent is exceptionally small.


  I though of using LDR's (Light Dependent Resistors)


As others have mentioned, LDRs are very slow and aren't used often for optical communications.