Use ssh. Once yo have exchanged the keys yo don't need a password. Read the ssh docmentation to get the concept.
Test the ssh connection by issuing "ssh user@remote_server" - if this asks for a password you have done something wrong.
Now start your script by issuing "ssh user@remote_server command". This will be analoguous to rsh/remsh/rexec.
If you want to minimize any risk you might want to use this to check the connectivity before trying to remotely execute a command:
if [ f_CheckConnectivity remote.server user ] ; then
ssh user@remote.server command
else
print - "Error: no authorization at or connection to host remote.server"
fi