How can I upgrade a sub-panel without upgrading the three-conductor supply?
Assuming your conduit is continuous from your basement to your subpanel, your best option is to use it as your ground path.  Metal conduit is an excellent ground conductor, and can easily handle hundreds of amps, even when a little rusty.  

Buy a conduit ground clamp, such as this one:


(Image courtesy of Home Depot)

Then, run an appropriately sized green or bare copper wire from the end of the conduit in your basement to your service panel, or any point along your grounding electrode system.  This will complete the ground path from your subpanel back to your main panel.  

Now, to complete the conversion, shut off the breaker feeding your subpanel, and remove the neutral-ground bond that's likely in there.  This may require moving neutral and ground wires onto separate buses if they're currently mixed.  

If you're not sure the conduit is continuous (i.e. it might transition to plasitc somewhere underground), you can now test that with a multimeter and an extension cord:  Plug it into any regular outlet in the house, walk to the subpanel in the garage, and test for continuity between the round ground pin of that extension cord and the metal conduit going into your subpanel.  If it has continuity, you're good to go.  If not, now pull another ground wire through the conduit and connect it at the subpanel and main panel/grounding electrode system (but still leave the conduit clamp connected as well -- it's good to bond conduit to ground even if it isn't serving as a ground path).