How to setup incoming traffic failover using two different Internet providers public IP addresses?

I have two separate Internet connections with two different providers “A” & “B”. All outbound traffic is setup to go out on either provider connections “A” or “B”. However, all inbound traffic comes in on one connection “A”.  This is because the DNS service (hosted by GoDaddy) A records points to webservers and a mail server public IP addresses provided by Internet provider A on connection A on our Firewall which then is directed to internal IP addresses of those servers.
Internet connection “B” also has Public IP addresses provided by the Internet provider “B”.
I want to make it so if connection “A” goes down, incoming traffic would come through “B”, or vice versa.  I can easily make the connection “B” provided public IP addresses point to internal servers but I am not sure how to setup DNS so if one is down it routs automatically to our other public IP addresses. Is this something I can do with our current DNS provider or do I need to use a different service? How do I set this up?