The 32 bit version of the jre is also known as the "client" runtime; the 64 bit version is called the "server" runtime. The server jre is often installed on servers but it will run find on a regular desktop, just as a 32 bit JDK will run fine on a server. It is just that the 32 bit version is not the best choice for a busy server as the 64 bit version generally has better performance (benchmarks can be found on the internet).
Now, there are plenty of java applications and libraries. Many of the libraries are included in the repositories. Some are "noarch", which means that they don't care whether you install a 32 bit or 64 bit jdk. Other ones are compiled to work with one or the other. The Eclipse IDE, for example, is available in 32 and 64 bit versions; if you match either with the wrong JRE, it will crash as soon as it is launched. If you install a 64 bit Linux OS, such libraries/applications will be 64 bit so it is better to install the 64 bit JDK if you want to make use of them - unless you fetch and install the 32 bit versions manually.