I don't know why Riot decided to call it ping but what we call ping is actually latency. Latency is the round-trip time for a packet of data. When you have 70 ping, it means that it takes 70 milliseconds for data to go from your computer to the server and for data to come back. Now, online games are designed to send as little data as possible per round-trip which is why you don't need much bandwidth. But if you're planning on having downloads running while playing then you should have a high bandwidth connection.
When you have downloads running it's going to affect your ping because as far as your internet is concerned, the return trip of your game's data is just another download. It has no reason to prioritize it over other downloads so you end up with situations where the round-trip time is greatly increased because your internet connection for the moment is downloading other data so your game has to wait for it's data. We experience this as lag or ping spikes. Now, with a high bandwidth connection, what usually happens is that whatever you're downloading won't be able to use all the bandwidth so even when downloading there is still enough bandwidth left fast round-trips of the game's data in between your downloads. Uploads will have the same effect because like downloads, your internet won't prioritize your game's upload over other uploads so that will affect round-trip times as well, only in this case your game is waiting to send instead of waiting to receive.