Jasper should work with all current versions of Java. Except under Windows the full development kit, not just the JRE, is needed. Unless indicated otherwise, you can download the JDK from https://java.sun.com/.
Jasper is built using a particular version of Java, as appropriate for each platform, but it is possible to recompile it to work with older versions as well. The version differs between platform, see https://sicstus.sics.se/ for details.
For some platforms, Jasper is only supported under the following conditions:
Using Jasper from Java may require that DYLD_LIBRARY_PATH be
set up so that Java can find the SICStus runtime library. That is, you
may need to set DYLD_LIBRARY_PATH to the location of the SICStus
runtime libsprt4-5-0.dylib.
On OS X, embedding Java 7 or later in SICStus, e.g. by using
library(jasper), may fail, possibly with a dialog about the
need for a Java SE 6 runtime. This seems to be caused by missing
information in the section ‘JVMCapabilities’ of the file
Info.plist in the Java installation. As of December 2017 there
was no official information from Oracle about this but several
suggested workaround can be found on the net.