Actual plots of growth hormone secretion patterns, with age on the horizontal axes, tell a different story. See, for example, the graphs below, from professionalmuscle.com. They match the graphs one sees in empirical academic papers. The graphs below (click to enlarge) are particularly good at highlighting some interesting patterns of variation.
On the left side, bar charts show secretion patterns grouped by age ranges during a 24 h period (at the top), during wake time (at the middle), and during sleep (at the bottom). On the right side is the actual data used to build the bar charts. As you can see from the graphs on the right side, the drop in growth hormone secretion follows a pattern that looks a lot more like an exponential decay than a linear pattern.
The drop is very steep from 15 to 40 years of age, after which it shows some fluctuations, going up and down. Interestingly, people in their 50s and 60s, at least in this dataset, have on average higher growth hormone levels than people in their 40s. Of course this may be due to sample bias, but the graphs suggest that there is a major drop in growth hormone secretion, on average, around age 45.
As you can see, there is a lot of individual variation in growth hormone levels. If you look carefully at the graph on the top-right corner, you will see a 50 year old who has a higher 24 h growth hormone secretion than many folks in 15-30 age range. This pattern of individual variation is common for the vast majority of traits anyway, and often the distribution of traits follows a normal, or bell-shaped, distribution. The bell-shaped distribution becomes clear when the traits are plotted based on frequency.
Growth hormone is secreted in pulses. In case you are wondering, growth hormone secretion in young women is higher than in young men. See the graphs below (click to enlarge), from this excellent article on growth hormone by Cummings and Merrian.
Yet, women do not put on a lot of muscle mass in response to weight training, regardless of the age at which they do weight training. This means that growth hormone, by itself, does not lead to significant gains in muscle mass. Androgenic hormones, like testosterone, play a key moderator role here. Muscle mass gain is the result of a number of things, including the combined action of various hormones. To complicate things further, not only do these hormones act together in an additive fashion, but they also influence each other.
Another reasonable conclusion from the data above on growth hormone secretion in young women and men is that growth hormone must indeed have major health-promoting effects, as most of the empirical data suggests. The reason is that, from an evolutionary standpoint, young (or pre-menopausal) women have always been the evolutionary bottleneck of any population of ancestral hominids. High survival rates among young women were a lot more important than high survival rates among men in general, in terms of the chances of survival of any population of ancestral hominids.
Higher survival rates among young ancestral women may have been enabled by higher levels of growth hormone, among other things. The onset of the metabolic syndrome, which is frequently in modern humans around age 45, may also be strongly influenced by falling growth hormone levels.
How can growth hormone secretion be increased after age 45? One obvious option is vigorous exercise, particularly resistance exercise.