The space that would be used by a CRC32+size gives you enough room for a bigger CRC which would be a much better choice. If you are not worried about malicious collision that's it in which case Thomas' answer applies.
You didn't specify a language but for example in C++ you got Boost CRC giving you CRC of the size you want (or you can afford to store).
CRC is most an error detection method than a serious hash function. It helps in identify corrupting files rather than uniquely identify them. So your choice should be between MD5 and SHA1.
If you don't have strong security needings you can choose MD5 that should be faster. (remember that MD5 is vulnerable to collision attacks). If you need more security you better use SHA1 or even SHA2 .
CRC-32 is not good enough; it is trivial to build collisions, i.e. two files (of the same length if you wish it so) which have the same CRC-32. Even in the absence of a malicious attacker, collisions will happen randomly once you have about 65000 distinct files with the same length.
A hash function is designed to avoid collisions. With MD5 or SHA-1, you will not get random collisions. If your setup is security-related (i.e. there is someone, somewhere, who may actively try to create collisions), then you need a secure hash function. MD5 is not secure anymore (creating collisions with MD5 is easy) and SHA-1 is somewhat weak in that respect (no actual collisions were computed, but a method for creating one is known and, while expensive, it is much less expensive than what it ought to be). The usual recommendation is to use SHA-256 or SHA-512 (SHA-256 is enough for security; SHA-512 may be a tad faster on big, 64-bit systems, but file reading bandwidth will be more limitating than hashing speed).
Note: when using a cryptographic hash function, there is no need to store and compare the file lengths; the hash is sufficient to disambiguate files.
In a non-security setup (i.e. you only fear random collisions), then MD4 can be used. It is thoroughly "broken" as a cryptographic hash function, but it still is a very good checksum, and it is really fast (on some ARM-based platforms, it is even faster than CRC-32, for a much better resistance to random collisions). Basically, you should not use MD5: if you have security issues, then MD5 must not be used (it is broken; use SHA-256); and if you do not have security issues then MD4 is faster than MD5.
