API Gateway deployments are generally pay per request, so deploying to multiple regions shouldn't cost anything extra. If your back end is lambda that's also pay per use, but if the back end is EC2 or similar that can be more expensive to have multiple regions.
My initial design wouldn't include multi-region unless you've already experienced multiple failures in a single region or you need extremely high reliability - four or five nines. Human error is more likely to break your systems than AWS outages.
API gateways are redundant and reliable, and while AWS does fail it's not all that often, and when it does so many websites are down you're in good company. Not ideal if you have a tight SLA though.
If I did need high availability / multiple regions I would deploy the gateway and workload to multiple regions, then first look to use Route53 to provide a single URL for those those APIs. With R53 you can use any routing you like geolocation, latency, or failover. If the back end is expensive you could look at a cold or pilot light deployment model with failover routing, if performance or availability was more important you can deploy to each region and use geo / latency based routing.