We will be visiting the Dubrovnik port while on a cruise in May. I have contacted a taxi group and they can drive to Montenegro as it is 2 hours away. My question is-Is it too risky to go across a border to another country while on a one day stop in port?
It is probably not risky, but figure it will take up nearly your whole day. Figure 4 hours or so travel a couple hours there...and the day is pretty well gone. If I were to rate Kotor, the Bay and area are nice and the town is interesting, but compared to Dubrovnik, Dubrovnik wins. Why not just spend the day in Dubrovnik, spending a little time on a beach as well as in the town?
I'm with Paul. Kotor is nice, but the Dubrovnik area is a much better place to focus your limited time. There are boats that leave the old harbor within the walled city that take you a short ways (10 miles) down the coast to Cavtat which is a wonderful little town. It's a great spot to spend some of your day with a nice seaside promenade, Roman ruins, and outdoor restaurants along the water to sit and soak it all in. You can also catch a boat from the old harbor over to the island of Lokrum which is a nice place to walk around in a natural, historical setting and see exotic plants from around the world in the botanical garden. The walled city itself deserves a number of hours, so I wouldn't recommend Kotor with just one day available to you.