Will an 8gb DDR2 desktop run games like Minecraft smoothly?

RAM doesn't matter too much. What matters the most is your graphics card.

Probably not, here's why. DDR2 RAM hasn't been commonly used in around 7 years or so. Which means the rest of the computer that the RAM goes to is going to be basically obsolete by today's standards.

Minecraft isn't a particularly demanding game, It's more CPU intensive than graphics card heavy. A fast dual core and mild video card ( GT 640/740 or Radeon 7770/R7 250X ) is all you need for vanilla minecraft, but once you start adding mods (and you will), the hardware requirements go up significantly. For a moderate number of mods an FX6300, i3 or i5 with a GTX 750 ti or Radeon R9 270 would be sufficient. If you're running a lot of mods an i5 with a GTX 960 will do.

What's the processor and graphics? Minecraft can be quite demanding on the processor. Stock it doesn't require much graphics performance but if you get into visual mods that changes.

Wit that amount of ram you will be OK.
The processor is important. I would use a duel core or the new i3. The stock graphics on an i3 is fine.
I run on 2gb of ram (but i use a puppy linux install which only needs 56mb of ram and 2% cpu on an i3 so there's loads left for minecraft to run. Unfortunatly windows requires 1gb ram to run and even more ram to run graphics.

So the answer is yes ram is OK if you have a intel duel core or i3 or i7.( an old pentium p4ht 3.6ghz would do at a pinch.