Blue LED always on with led light strip
You will need a ohmmeter. Disconnect the IR Controller (the white box with white wires). If you check the the voltage drop (in the connector of the white wire) between the Positive (+) and the different wires. If you find that they all respond to commands from the remote... except for the port for the blue wire (which controls the blue light), then the problem might be with the remote or the receiver. Again, if the "blue port" always reads 12VDC (despite trying to turn it off), then the problem is with either the remote or the receiver (and not the strip). 

However if it does turn on and off upon command, then there must be a short in the strip or connectors (powering the blue wires). 

If you look carefully along the strip you might be able to see where the short is (often looks like a burned/discolored spot (or a discolored LED). If you bought the non-waterproof strip, then you might find some debris or dried soda or something on the strip. Or if you have the waterproof strip, you might see a crack in it where humidity has collected and is causing a short between + and B.

If everything seems to be looking okay, then you could get some short pieces of wire (strip the insulation off of both ends) and insert them into the connector- making sure to only connect one wire to each port- and perhaps diagnose which wire is connected to the Blue lights by connecting one wire at a time... then two wires at a time... and (if necessary) three wire combos. Unfortunately, this may not be a fruitful piece of information, because it seems that the problem is with the strip or one of the LEDs anyway, and if you don't see any obvious faults then there's not much you can do figure out where the issue is on the strip.

So of course, if you have no idea where the fault lies (just on the strip somewhere) you'll have to cut out the bad section by cutting the strip in half. One half of the strip should be okay... then you cut the bad half in half... and just walk-it-down until you're satisfied. Then you can reconnect some of the lengths worth keeping.