How to check your oracle is 32 bit or 64 bit?

1. You can retrieve this information from the data dictionary view v$version. Banner will tell you 64-bit if the 64-bit version is installed.If it does not specify 64-bit then it is 32-bit (although it does not explicitly say so). Same information you get when login into SQL*PLUS.
2. On Unix, the file command is also available . Example:
$ file `which oracle`
/oracle/home/oracle/product/8.1.6/bin/oracle: ELF-64 executable object file
- PA-RISC 2.0 (LP64)
Or
/opt/oracle/product/8.1.7/bin/oracle: ELF 32-bit MSB executable SPARC
Version 1, dynamically linked, not stripped
3.There is another way of finding out via sqlplus command if you do not have access to the machine itself.
select distinct address from v$sql where rownum<2; ADDRESS ---------------- 0000000386D3E060 SQL>
If the address is 16 characters long, it is 64 bit.
If it is 32 bit you will get an 8 character address.
SQL> select distinct address from v$sql where rownum<2;
ADDRESS
--------
826A774C