How to drill holes in 3/8" thick steel plates?
3/8" mild (hot-rolled) steel isn't difficult to drill, but any bit will fail if you get it hot enough to melt the cutting edge. Each hole should take no more than a minute.


Use a sequence of sizes (1/8", 1/4", 3/8"). This makes for quicker drilling and allows each bit to cool between uses. If you only have "pilot point" bits on hand, use a starter bit that's no larger than the pilot on the next bit to keep things centered.
Use a lubricant. Common household oil is probably adequate.
Keep the bit in alignment. Binding generates heat much more quickly than the cutting process itself.  
Cool the bits while they're not being used. A fan across them would do well, or lay them on a damp rag.
Keep the bits sharp. Dull bits create more heat (in the bit and in the operator). A bit sharpener is a great investment for any DIYer, but even a bit of emery cloth can help touch up cutting edges. 


More tips