Is there anything I can do to prevent my grandmother from dropping her pills and forgetting them?

I would have her pharmacy package her pills in blister packs, this helps the patient know which pills to take and it helps their family/caregivers keep track of any missed doses. Tell her that it is the pharmacy's new policy for all their seniors.
Setting her pills on a tray may also help so that any fumbled pills will land there instead of on the floor.

I agree with cwillie. My husband takes 12 pills a day. Our visiting nurse suggested we sign up with a company (Exact Care) who prepackages the daily dosage of pills in a plastic bag. All he has to do is cut or tear the top and tip them into his mouth. There is no extra charge for this service and it works!

Do you know what your grandmother's primary ailment is? Are you able to discuss the problem with her doctor? Does the problem seem to be equal between manual dexterity, poor vision and poor memory? I'd try to figure out what the primary problem is. If she has cognitive decline or dementia, then it wouldn't surprise me that she is forgetting to take her medication.

And she may be refusing to use the pill organizer, because it may confuse her. When there is cognitive decline, it's really not possible for them to learn new things. The pill organizer may be just too overwhelming for her. 

Sometimes, depending on the condition that is causing the problem, it's just up to the caregiver to supervise them taking their meds. We may not be able to rely on them to remember or do it correctly, nor to correctly report what they took. Is there someone available to do that? There is also the danger that they forget what they took and take too much medication.

I don't recommend promoting the use of the pill organizer. I am 86 years old. Although I don't think I lack manual dexterity, the pill organizers I have bought all have front surfaces that slant inward slightly, making it difficult for me to grasp the pills. Also, I find it unpleasant to see and differentiate the different white pills. I'm not suggesting you do what I do, but just want to mention it. I remove each pill from its original container and take it. Then I turn the container upside down, so if the phone rings or there is some other distraction, I know that I've taken the pill. This method also alerts me to notice when I need to reorder pills. I recommend that you don't let your grandmother take her pills by herself.

With Mom I used a shot glass. She could swallow 4 pills at a time. Mom spits hers out now so the CNAs put her pills in applesauce.

You are right, using those small white paper cups [ketchup holders LOL] would be a great help. Buy some and see how that works. Order them on-line or see if any of the local drug stores carry them. Now, the big challenge is to get Grandma to use those cups.

As for the AM/PM pill organizer, as great as they are, your Grandmother will still drop the pills and claim they were from yesterday.... [sigh]. Hopefully there are no pets in the house that would get into those dropped pills.

I made my own "blister packs" when I went on vacation.
I took a "seal-a-meal" type food saver and made a large bag then turned it sideways and sealed long pockets. Then I pushed each days worth of pills down the sealed "tube" and when they were filled all the way across I sealed that then continued with the next row. Time consuming but I got 4 weeks worth of pills done and did not have to worry about doing it again for a month. I do not know what the pharmacy would charge to do the blister packs or if it would be covered by insurance.
Another option would be the little snack size zip top bags. They are smaller than the sandwich size.
I would be concerned about leaving pills in a little cup for several reasons.
Humidity is not good for some.
Light is not good for some medications.
The cats could fine little pills in a cup irresistible, for some reason my cats live to stick their paws into cups and bowls to play with whatever is there.
And the little cups are easy to tip over.

My hubby is a liver transplant patient--went from taking a multivitamin (sometimes) to being on several drugs. We use the AM/PM snap tight containers--most pharmacies carry them, they are pretty large and the writing on them is HUGE. Hubby sets out his pills for the week, snaps them shut and off he goes. (The snap tight thing doesn't last forever, so I buy new ones all the time....then I know they're not going to pop open in his pocket).

Now, to say he's 100% compliant is not true, as the MOST IMPORTANT drug he takes (his antirejection med) is so tiny--and it's the kind of capsule that gets "staticky" and he will often drop it as he throws back the mouthful of pills. I'll find those little buggers everywhere. Thank goodness I clean a lot---I have 13 grandkids and so far, they have all been great about bringing me pills if they find them.

As "independent" as Gramma wants to be, she needs help in this dept. Someone has to step up and have the "talk" with her that she needs a little help with the meds. If you separate them out in small packages, labeled and she can open the container and take them on time, great, she's still independent! If you HAVE to, get the little white cups and "administer" them, the way the hospital does. Stand there and watch her take them, but chit chat with her--don't make it a "I'm making you do this" kind of a thing.

A side note, that is funny/sad: My FIL was in the last stages of his life and on 3 meds. 2 gave him terrible side effects and he was tired of taking them. I'd go all the way out to his house and give him breakfast, his pills, then go back in the evening for dinner and evening meds. I'd been handing them to him, as he sat in his recliner. Only after he passed and I pulled the recliner out to clean, did I find a huge pile of pills on the floor. It didn't matter at that point, and they weren't keeping him alive, but it gave me a chuckle. I remember thinking "dad, you won. you hated those meds".

In retrospect, even though I GAVE him the pills, I never once watched to see if he actually took them.

Good suggestions above. I agree that grandmother may want to be very independent, but, sometimes when they have issues that create a great risk to their safety, only they can't see it. It may be that this is just something that has to be supervised.

I don't know if this will help. I use the pill organizers when I travel, personally, I find them hard to use. One pill doesn't come out easy. I use a small lazy susan makeup table. I have the morning one together, and the evening ones together. If the morning ones are in front, I know I took them.

When I take the pills out of the bottle, I put them on a dark hand towel. That keeps them from rolling around and the dark towel helps me see the pills.