The best thing you can do, is to forget about trying to consciously tense your abs during daily activities.
There is nothing inherently wrong with relaxing the stomach muscles, especially during normal activities. For example, consciously keeping your abdomen pulled in all the time can restrict basic bodily functions like breathing or digestion.
If you want to improve 'core strength', forget about sitting on a yoga ball and wearing ab belt for extended periods during the day. Rather, focus on living an active life as much as possible, and walk, lift, run, carry, climb, and throw as much as you can - they all involve your core.
The abdominal region is very complex, it moves and stabilizes the spine during a wide range of activites: standing, walking, running, lifting, climbing, throwing and so on. The abdominal muscles must be responsive to these challenges, each activity needs a different pattern of tension-relaxation. Therefore, the absolute worst thing you can do is to put a belt around your abdomen during the day, and think it will make you strong. Quite the opposite: it will make you weak and rigid at the same time.
There are many fitness exercises which are very effective at training your core. In general, exercise is more effective for the core if it involves more muscles around the trunk, not only the abdominal muscles. (ref) That means variations of planks (mountain climber planks, side hovers), as well as strength training (squats, deadlifts, pullups, presses, loaded carries) will all be great exercises for the core. For more tips, there are also many great answers on this site, see this, this, this, and others.