Can I put drop handlebars on my commuter bike?
It's very unlikely that your brake levers can be modified to increase the diameter enough to handle the change from 22.2mm to 23.8mm.  The brake clamps just aren't adjustable like that, typically.

Where on the handlebars were you thinking to mount the brake levers? If you were thinking at the middle (next to the stem), then you're likely in the area where the handlebar is wider (the "clamp diameter") which is even worse.

One option is going with "trekking" (aka "butterfly") handlebars. These are typically the same diameter as straight bars and can easily accomodate the same brake/shift levers.  They look a little funny, but do give you more handlebar positions similar to what drop bars do.

Or just add bar ends to your existing bars. The right bar ends can give you a comfortable sideways hand position similar to the main hand position people use on drop bars.

Another thing to consider is handlebar positions on the drop bars.  Typically the position people use most often is on the brake hoods. Brake levers made for drop bars tend to have comfortably shaped hoods with a little bit of give in them. Brake levers made for flat handlebars don't generally have a comfortable place to rest your hand.

If you do decide to switch your brake levers, be aware that the amount of pull is likely different so you'll either have to be careful with which brakes you get or do something to accomodate the difference.

Here's a nice article that includes more details about changing to drop bars: http://sheldonbrown.com/deakins/handlebars.html