Who currently pays the miners for processing transactions? The graph: Bitcoin mining cost per transaction leaves an ambiguity. Who pays the transaction costs?
According to the list of available charts, that chart is showing "miners revenue divided by the number of transactions".  That is, it's the average amount a miner earns for each transaction they process.  "Bitcoin mining cost per transaction" isn't a very good title for the chart.

So far, the vast majority of miners' earnings comes from the 50 BTC per block rewards, with a tiny fraction coming from the transaction fees paid by the people creating transactions.

So to answer your question, nobody pays the vast majority of the cost; it is created out of thin air as the reward for mining a block.  You could see it that each new block generated inflates the money supply and so devalues all existing Bitcoins, in which case you could argue that everyone holding Bitcoin is paying via inflation.  But the inflation is predictable, and so already priced in.