Can you set your Coop Commanders back to level 1?

TL;DR Sure can (for level 15 commanders) as of a week ago . Just click "Activate Prestige" in customization menu to reset to level 1.

Well up until literally a week ago I would say the only way is to create a new account.

But with Patch 5.0 live, you now have the option to reset any level 15 commander back to level 1, though this feature was not added for the nostalgia of those pining to play nub commanders again (whatever floats your boat).

It comes with this new "Prestige" system, where each commander can activate 1 of 3 Prestige Talent options that give the commander distinct buffs and nerfs. For instance:

Raynor:

Prestige Level 1 "Backwater Sheriff": Biological combat units gain 100% additional life, but MULES are unavailable.

Prestige Level 2 "Rough Rider": Afterburners increase attack speed by 100% and its cooldown is reduced by 50%, but units no longer cost less, and the speed bonus is reduced by 50%.

Prestige Level 3 "Rebel Raider": The Starport no longer requires gas to build, and its units cost 30% less gas. Vikings, Banshees and Battlecruisers increase top bar ability cooldown rates by 1% per supply used, but all combat units cost 50% more minerals.

To activate these options, in the commander customization menu (before just used for adjusting mastery stats), click the "Activate Prestige" button on the top right. It will prompt you to make sure you really really (it prompts twice) want to do this, because doing it will drop the commander all the way back to level 1, just like you wanted.

And you can do it 3 times for each commander, one for each Prestige Talent.

Now if you just want to play the commanders vanilla, just don't select a Prestige Talent, click OK and queue up.

None of this will reset Mastery points, but you don't have to use all of those if you don't want to...