Risk of Drooping Eyelid for Crow's Feet Botox Injection?

This is actually an excellent question.
Of course the most common cause of drooping of the UPPER eyelid is inadvertent injection into the eye lifting muscle, or aggressive forehead Botox, causing the forehead to descend excessively.
However, when Botox is used for the crow's feet area, it can deactivate the orbicularis muscle. this can cause drooping of a borderline weak lower eyelid, or exacerbate cheek droop that a patient has, causing LOWER eyelid drooping. To understand the effect on each of the muscles on cheek and lower eyelid positioning, we have some literature below.

An expert injector with many years' experience treating crow's feet with Botox should be able to give you an excellent result with no eyelid or forehead droop. Drooping is caused by overinjection or by injecting the wrong muscles. In either case, it's a mistake that a professional should be able to easily avoid. Please consult a board-certified dermatologist with expertise in Botox injection.

Short answer -- you are unlikely to receive any eye-LID or eye-BROW drooping if you only receive Botox in your crow's feet...Your question does, however, warrant discussion of two rare, but known, complications of Botox...(1) A drooping eye-BROW may happen when the brow-elevating muscle in the forehead, the Frontalis, receives too high a dose of Botox, or alternatively, if the Botox is sub-optimally placed too low in the forehead. Of note, it may also happen if you have a low set eyebrow to begin with, in which case any Botox to the Frontalis increases the likelihood of a brow droop. A droopy eyebrow from Botox can sometimes be improved with Botox injected into the outside part of the eye (the lateral aspect of the orbicularis oculi muscle) to generate a bit of a brow lift in that area -- by injecting more Botox and paralyzing the orbicularis muscle that normally acts to depress the brow in that area, you may get a slight compensatory brow lift... (2) A droopy eye-LID may occur if the Botox is injected too close to your eyelid-elevating muscle, the levator palpebra superioris. In such a scenario, the Botox will diffuse inadvertently onto the levator muscle and cause an eyelid droop. You may have an increased risk of eye-LID drooping if you have a weakened upper eyelid muscle for neurological reasons, or a deeply set eye-BROW that would be more prone to drooping (as discussed above) and result in skin gathering over the eyelid making the eyelid appear like it was drooping. A droopy eye-LID due to Botox can be treated with Apraclonidine eye-drops which can provide a small (2mm) improvement -- Apraclonidine 0.5, 1-2 drops, 3 times per day. Make sure you put in one drop at a time, tilt your head back, and close your eyes to make sure none of the eye-drop leaks out. Be sure your prescribing physician discusses all the potential side-effects of the drops, such as "adrenaline-like" symptoms like anxiety or heart pounding; you may also experience eye irritation, eye dryness, and eye pain, amongst other symptoms. If these symptoms occur, you will likely need to take some lubricating eye drops, lower the dose, switch the eye-drops, or stop the drops altogether...My recommendations are to seek the services of an experienced physician injector. I think the key with Botox lies in truly understanding the anatomy of the injected area, and more importantly the variability in the anatomy between patients -- for brows, the forehead, and anywhere else you plan on receiving a Botox injection. This includes having a firm understanding of the origin, insertion, and action of each muscle that will be injected, the thickness of each muscle targeted, and the patient variability therein. As an aesthetic-trained plastic surgeon, I am intrinsically biased since I operate in the area for browlifts and facelifts, and have a unique perspective to the muscle anatomy since I commonly dissect under the skin and see the actual muscles themselves. For me, this helps guide where to inject and where not to. However, with that said, I know many Dermatologists who know the anatomy well despite not operating in that area, and get great results.Good luck.Dr Markarian

Generally eyelid droop comes from injections into the forehead above the brow.  That's not to say it's impossible for what you are describing to have happen, but it would be very unusual. 

It would be more common to have a problem with saggy under eyes when injecting the crow's feet poorly, so the risk is more below the eye than above it with this type of injection.

 Botox is my first choice for treating crow's feet, which are caused by the dynamic muscles around the eyes and the muscles that allow you to smile: the orbicularis oculi, levator labii superioris, and the zygomaticus major and minor. A board-certified dermatologist with expert injection technique will only administer Botox in those muscles and will only inject enough to relax the muscles, not freeze them. Drooping eyes are the result of poor knowledge of anatomy and/or poor injection technique. Please do your research and find a board-certified dermatologist with expertise in using Botox to rejuvenate the face and eyes.