There's two ways to deal with the botting issue.
- Detection of bots, delayed bans and banwaves. Just like with any other third party applications Riot has an anti-cheat that is utilized to detect the injection of "scripts" which alter the game files. Some bot programs are a little more difficult to detect as a few of them actually utilize different methods to achieve their goals, but nevertheless it is an endless cat-and-mouse game so to speak.
This method is not particularly efficient. Imagine a hole in the side of a ship and Riot has developed some pumping mechanisms to keep the water level relatively low. This doesn't fix the hole in our analogy but it is doing SOMETHING.
- Getting to the heart of the issue - why people bot and how to curtail the incentive to do it.
This is far more in-depth but the basic reason that people bot is to get to level 30 because the leveling process is a nightmare. Bots do significant harm to low MMR ARAM games as well as those in Co-Op Vs. AI and even a few low level summoner's rift games. They also further damage the new player experience.
Why do people bot? It's an industry that makes a lot of money! Selling level 30 accounts is a pretty big deal. Some people have made thousands, if not tens of thousands of dollars selling botted accounts. Smurfs would rather buy an account than level from 1-30. Players that are banned would rathr buy an account than level and herein lies the problem. If there was no longer a leveling process from 1-30, but instead a requirement of a minimum number of wins in normal draft mode, this could significantly curb the bot issue.
There would no longer be a need to sell accounts because the leveling process would be gone. Bots wouldn't be able to play normal draft and, while there may still be some, the need to buy accounts would be less so than it is now. No matter what solution you come up with there will always be bots, but what I'm proposing would reduce the problem significantly.
To go along with this I would suggest creating a much better new player tutorial with a beginner and intermediate mode where the player learns about the basics of last hitting, lanes, summoner spells and items.
Riot is likely hesitant to do this but I think the upside far outweighs the negatives