Checksums are as reliable as there are combinations of bits in them. The real answer depends on what the serial data contains and whether you can afford to add any correction to it. CRC isn't difficult and is more reliable than a simple checksum.
If you have a high error rate and not randomly distributed errors, e.g. errors associated with specific data words, crc is better. For low errors rates, the performance is similar.
From your description it sounds like you are in control of whatever means you choose (the protocol) in which case I suggest you use the simple expedient of transmitting each byte twice and the receiver comparing them, ignoring those that don't match.
I use this in lots of low power simple communications with simple micro's, it is neither memory nor MIPS hungry