The general rule of thumb is yes, 32bit apps will work on 64bit operating systems. The common exception to this rule is when the application itself has "hooks" to the kernel or needs certain drivers to run - most famously VM softwares - and then it is better to get the native 64bit version.
Also, plug-ins such as Java, in order for it to be compatible with Internet Explorer 64bit, should be in 64bit too.
That said, if you are running a 64bit Windows - always try to find the 64bit version of the app. If none exists, then use the 32bit version.