There was a lot of panic and speculation leading up to the first block reward halving in late 2012 (from 50 BTC to 25 BTC). However the actual incident turned out to be uneventful. The next halving to approximately 12.5 BTC will will happen in 2 years and will probably be just as uneventful. In about 140 years, if Bitcoin is still around, the fees will be the majority of miner's income. I bet nobody will notice the block reward disappearing (unless 1 Satoshi is worth a fortune then).
A few minutes of research shows that in the past few weeks the block reward is generally 25btc + ~0.2btc in fees. (around 1%).
If bitcoin grows in popularity over the next 6 years, the 2020 halving will reduce block reward to 6.25btc + ~0.8btc fees (4x the fees of today due to much higher transaction volume). That's now over 15% of the miner incentive coming from transaction fees.
Of course this is a lot of speculation regarding volume, but it shows the idea of fees eventually funding the miners
