They don't want to put them in a kennel, leave them at home for a long period of time under someone else's care, or are just afraid to try it. They should review this decision again. Try asking the following questions about YOUR pet and see if it helps you travel more often. If you answer Yes to either of the above, you might try a short trip. If the pet doesn't seem to stressed, try a longer one.
Some pets travel best in a carrier, others want to travel loose.
It's really safer in a cage, strapped down to the floor. There are several ways to do this. The most important one is keep your pet on a leash at all times when you are in a strange environment. This DOES apply to cats, too. My cat is in a figure eight harness at all times, with his tags attached, and on a lead whenever he is out of the motorhome.