My opinion:
Not really a huge cause for concern. Stuff will brown and drop off, and the cutting will probably be fine. This usually happens in a very humid environment when the top growth outpaces root growth. If the same happens in a less humid environment, then you might not get the top growth, or it will shrivel before it gets brown.
Most cuttings will do a bit of this at some point. Important thing is to keep giving them fresh air periodically, and don't overwater. If the mix becomes too moist, that can rot the roots and spell death for the cutting. The top is much less important than what is going on below the soil.
No need to use fertilizer until much later in the process. Very diluted fertilizer probably won't hurt, but won't help much either. Think of it as a seed. For the first weeks/month or whatever, the seed will use its stored energy to produce leaves/roots. Cutting is similar. Only once it has established sufficient roots to be able to provide moisture to leaves for significant photosynthesis will it need any fertilizer.
Ideal situation is that roots grow ahead of leaves, because without roots, leaves really can't do anything useful. But it doesn't happen this way in practice. Leaves might grow first. These leaves aren't really that useful. If you can save them, fine, if not, if they drop off, no big deal, as long as roots are still developing. I don't worry about brown stuff on the new shoots. I get worried if it happens down below.