My guess would be post-WWII. An old zine I have confirms its presence in the early 1990s. So that "pinpoints" construction from the 1940s to the 1980s. Regardless, its old, and the wooden planks that make up its floorboards squeak and shake and flex a bit.
Lafayette bridge - looking west.
Lafayette footbridge viewed from the south.
Lafayette pedestrian bridge - looking east.
I feel it worth mentioning that this bridge doesn't actually have a name. "Lafayette footbridge" or "Lafayette ped[estrian] bridge" refer only to the fact that it abuts S.E. Lafayette street. I don't think the Marquis de Lafayette ever set foot in Portland, Oregon, much less the Brooklyn train yards.
The Brooklyn-Gideon Footbridge, by contrast, is clearly a more recent effort (although also officially unnamed). In fact, I swear some online maps still show an actual street RR crossing here, a supposition supported by the dead-ending and concrete blocking-off of Brooklyn Street.
Brooklyn-Gideon footbridge, viewed from the east.
Since the Brooklyn-Gideon Bridge doesn't span the Brooklyn Yards proper the way the Lafayette bridge does, hanging out at Brokklyn-Gideon will bear witness to many more people using the tracks as a shortcut through or around the mess that is the confluence of Powell, the train tracks, 99E, Milwaukie, Division, and SE 11th & 12th Avenues. Train-hopping can also be observed from here.