I was thinking of cleaning up the seam and soldering it with regular plumbing solder or use some of the sticks of silver solder that we use in A/C systems. Will this work? what kind of solder should I use?
Silver solder used in AC systems is commonly silfloss or 15% silver. This will not work on your radiator! It melts at a much higher temperature (about 1450 degrees) than the solder used on your radiator.
A good choice is the original solder material that was used. It is simply soft solder or lead solder. You can get this a roofing supply companies for gutter work.
A better choice might be Silvabrite 100 which is a stronger lead-free material. it is a direct replacement for lead solder and was put on the market to replace lead solder in domestic water piping. It's readily available and is easy to work with. It is also a stronger metal, so the tank might not split as easily. It is only better in this case because it is stronger, not because it is lead-free. It is a bit harder to fill voids with than soft lead solder because it is less viscous when liquid. This makes it harder to fix radiators with loose fitting pans. You can't stack it as easily.
Lead solder is good because it melts at the same temp as the rest of the solder in your radiator instead of higher. Higher requires careful heat control so you don't melt the solder holding the tube assembly into the lower pan.
Don't try to fix any cracks with soft solders by just soldering over them. It won't hold the pressure for long.
There's more to the story than simply saying "Silver". There's a huge difference on the material strength and the melting temperature. Even Silvabrite has .4%. You need solder for water piping, either soft lead or lead-free. Lead-free has trace amounts of silver. But is not silver solder, per se.
A good water base flux like C-Flux works very well. Lots of flux applied with an acid brush as you go, lots of wire brushing with flux on the surface. Careful heating and dipping the solder in the flux before applying it to the surface will help. But heat control is the most important. Just enough to melt the solder in the area you are working and make it alloy with the soft solder already there. It's a skill you can work on and should before doing your radiator.