So, this is a tricky question. Most people answering this are giving technically excellent answers - distribute your zones across multiple nameservers, use high TTLs, invest in anycast, etc. - but I want to give you a contrarian viewpoint.
DNS is critical to the functioning of the internet. Everyone is on edge right now because of the recent DDoS against Dyn but this fear will fade.
I would like to point out that:
- This is the first major DNS outage (that lasted more than ~15 minutes) that I'm aware of in decades
- It was not a total outage (DYN faced the largest loss of service on the east coast, but was not globally down)
And also point out that every DNS provider on earth is focused on hardening themselves to DDoS right now.
Here's a funny but relevant tangent: in ~2004 some tiny site (fido.net?) with its own ASN broadcast a bad BGP prefix that cascaded to take down most of the internet core routing. Cisco and the major players fixed the bug, and we've never seen an internet wide outage due to a bad BGP prefix being broadcast again.
What I'm trying to say is - the entire internet relies on DNS. This DDoS against Dyn last week was extraordinary - in its magnitude, severity, and improbability.
This is not cheap or easy to mitigate from your end without owning your own infrastructure (which I can assure you is not cheap if Dyn's didn't stand up). The point of the DNS system is it is just supposed to work.
Which is my long way of saying you have an extremely valid fear that's so improbable and unlikely to ever happen again that you should just move on and not worry about it. Not because you're not right to worry and there's a 0% chance of this ever happening again (it might!), but because you have far more real and salient things that will affect your business in the near future that are better uses of your time, money, and effort than attempting to mitigate this.