It's a challenge to grow plants in containers without drainage holes. Drenching is the way we flush out all of the fertilizer salts, and flush IN a fresh supply of oxygen. It's a bit easier if the container is terra cotta so that moisture can evaporate through the sides as well as the top...and if the potting medium is especially porous it can be successful.

A challenge, but not impossible.

It's a recipe for failure, I'm afraid. The drainage problem is the least of your worries! All those herbs are large herbs - far too large for a window box. Some have an optimal size almost as large as yourself. You try squeezing yourself and several of your friends into a windowbox, eh? Can't be done.

Except for the mint, all those herbs except mint really hate wet feet. Roots sitting in wet soil will rot and be very susceptible to diseases, especially fungi. The soil itself will become diseased and sour.

I predict the mint will survive and choke the rest to death.

Do yourself and your plants a favour - start again, this time with plenty of drainage holes in the pot.

Thanks for the advice.
Do you think it would work if I dug them all out and potted them individually?

You could take a power drill and drill upwards to create holes. Besides the space problem, how much direct sun do your plants get? All the herbs you picked need at least 6 hours of direct sun -- will they get that on a balcony with northish exposure?

The basil is an annual, the rest are perennials, though the rosemary may not be perennial depending on your zone. Overwintering plants in a container on a balcony can be a challenge as well. This year, and being planted this late, they may stay small enough for the balcony box but will each need a 12" to 18" container for the long haul. And you want to isolate the mints in their own containers as they will overrun the others.

Good luck, happy cooking and welcome to herb gardening.

Thanks to all of you for your advice!

I bought a bunch of new pots (with drainage holes) and potted each of the herbs in their own large pots.

I also drilled drainage holes in the windowboxes and planted petunias, sweet potato vines, and fanflowers. Hopefully they'll do well there!

As for the sun issue for the herbs, I get indirect sun all day, and about 5 hours of direct sun in the afternoon/evening. I pushed all of the pots up against the balcony railing to maximize light, and I am hoping for the best at this point!

This is a great learning experience! Thanks again :)