The question of what is the proper dumbbell weight for an older man doesn't have an easy answer. If you've never exercised, haven't done it in years or have a physical impairment, such as arthritis, you may have to start at a very low weight. On the other hand, if you've been heavy lifting most of your life, either through exercise or your work, you could very well lift more than a teenager just starting out. The important thing is to choose a weight heavy enough to challenge you without causing injury and to progressively challenge yourself with more repetitions, weight and different movements.
Sarcopenia, now generally defined in the medical community as a loss of muscle mass, muscle strength and physical performance, can begin as early as your 40s, but it really kicks in around your 60s and 70s. Because it can occur even in physically active adults, it appears that other aspects of aging, specifically nutritional and neurological, may also be factors. That said, as a review of literature on the topic that appeared in "The Journal of Nutrition, Health and Aging" in 2009 reflects, sarcopenia can be slowed with physical activity, particularly resistance training, and the amount of loss may be tied the peak ability you achieved in earlier years.
The American College of Sports Medicine guidelines for slowing muscle loss in older adults is pretty much the same as for the general population. However, while at least two sets of eight to 12 repetitions is recommended for the general population, older adults may do better starting with one set. To choose the proper weight for one set, pick dumbbells with which you can do eight reps without breaking form but more than that would be hard. Proper form means slow and controlled motion, not locking your joints and keeping your head, back and pelvis aligned. So the proper beginning weight for biceps curls, for example, would be one with which you could do only eight or nine reps without the weight pulling your arm down -- causing you to lock your elbow -- or without arching or rounding your back on the lift.
It's not enough to pick one weight and work with it indefinitely. Progressive overload means constantly challenging your muscles by increasing the weight when reps become too easy. With a single-set program, as the original eight reps become easy, you would progress to doing nine, then 10 on up to 12. When you could do 12 in good form, you'd move on to the next weight. In addition, every four to six weeks you should change the exercise you do for that muscle group.
The 2009 literature review also noted that older adults may see more benefit from some workouts with heavier dumbbells and fewer reps. So you might do one week of resistance training sessions using 10-pound dumbbells for shoulder presses and doing eight to 12 repetitions. The next week you might switch off to using 15-pound dumbbells, doing only six to 10 repetitions.
If you're just starting your workout, err on the side of caution when choosing the dumbbell weight you start with. You can always add weight if it's too easy, but even one repetition with a weight that's too heavy could cause injury. You should also plan on staying with the same exercises for as long as eight weeks or however long it takes to start feeling and seeing the benefits -- if you stick with it, you'll know. Plan to engage in resistance training for all your major muscle groups at least twice per week with at least a 48-hour rest period between working the same muscles.