Curbs

[Bitmapped]

In some states (Ohio and West Virginia immediately spring to mind), it was common for bridges built during the 1950s through the 1970s on the Interstate system to be build with curbs on one or both sides of the roadway. Here's a classic example from I-70 in Washington County, Maryland: https://maps.app.goo.gl/HQpun3Xw6a2YNmTw8

Over time, in Ohio and West Virginia at least, it was common for guardrail to be extended across the bridge and built out to block off the curb like this example from I-77 in northern Ohio: https://maps.app.goo.gl/cFnkh2Lr6rVytBAW9

When the bridges are rehabbed, the curbs are generally eliminated like happened circa 2011 with the I-77 example: https://maps.app.goo.gl/CvWx94BAaZBeUqRs8

This brings me to my question: What was the point in having the curb in the first place? It seems that it was later considered a safety hazard, so I'm trying to figure out why this feature existed at all.

[freebrickproductions]

ALDOT used to do the same, such as with these bridges on I-65, which have since had the "curbs" filled in to make them more Jersey Barrier-like.

I don't believe it was anything super intentional, most likely the states just using the same bridge designs that they were using for the other highways at the time for the interstates they were building. I suppose there's also the thought that with the narrow shoulders, the curbs might keep the drivers from worrying that they might hit their mirrors.

[triplemultiplex]

A curb on a bridge is usually an attempt to channel run-off to a particular spot.

[Dirt Roads]

^^^
Correct.  There's two reasons for constructing curbs on bridges: (1) diverting and controlling water runoff; and (2) they serve as a miniature guardrail that is quite effective against [narrow] angle impacts.  Note that the effectiveness of curbs as a guardrail greatly diminish as the distance to the edge of the travel lane increases (actually, the distance to the travel centerline, which adds half of the lane width).

Jersey barriers, however, provide both of these functions and also serve to provide some level of protection in collisions.  In the case of [narrow] angle impacts, Jersey barriers tend to deflect the tire motion back towards the travel lane and the angular design minimizes carbody damage.  In the case of other impacts, Jersey barriers create a three-stage impact to the vehicle bumper to reduce the initial and intermediate impact forces.  In today's world of crumple-zone bumper technology, this feature is less of an issue (but there are still quite a few cars and trucks with old-fashioned bumpers on the road).  Unlike curbs, Jersey barriers can be used where there is a significant shoulder width on the bridge. 

For modern railroad bridges, curbs serve a third purpose: (3) to control the lateral movement of the ballast (ergo, large gravel used to restrain railroad crossties).

As with all roadway designs (including railroads), there is a fair amount of hydrology involved with the engineering of curbs and Jersey barriers.  For bridges with adjacent pedestrian sidewalks, it is also important to design the height of any curb to minimize the potential of flooding of the sidewalk area.