Q: Hello, I need my  iMac OSX v10.5.8 Power PC G5 to remotely controlling my iMac OSX El Capitan v10.11.3 across the internet. They are remote from each other on different routers. Can a version of Apple ARD complish this or any other available software

Remote access via ARD gets discussed here on occasion.  ARD alone is not related to this, as this involves configuring the networks and the firewalls to allow the requisite remote access — and that's irrespective to the services such as ARD or file shares or whatever that are running over the connection — either by linking the networks via VPN or by port forwarding or by establishing a way to VPN or (less desirably) port-forward into the target network.

I've posted on this topic in various threads, so if you go to the search page and use the search string /remote firewall vpn/ or such and then restrict the search to my forum ID, you'll find several dozen remote-access related discussions.  Several that show up in a similar and more ARD-specific search are here, here and here.   Those three and the other replies elsewhere should give you some background, some search terms, and some ideas.   My general preference is a firewall box that provides an embedded VPN server, or — if you're doing ad-hoc connections — one of the screen-sharing or screen-conferencing web sites if you are not in a position to set up your own screen sharing services on a site you're managing.

Have you considered using logmein? You would need to upgrade to at least 10.6 to use it. But it would allow you to remote in. No complicated configuration.

While you have to maintain local security regardless, some folks will have considerations and concerns around using services such as LogMeIn and TeamViewer, as you're ceding control to your systems to an entity outside your organization; to another (external) entity with the potential for access and control.  There are press reports and user reports that TeamViewer is currently the subject of various breaches reportedly related to the product, as an example of these concerns.

Any form of remote management has risks.

Including port forwarding wan to lan VNC traffic. VPNs as well. Though VPN is more secure then naked VNC traffic.