You can use ArcPy. A new polyline feature class will be created with the connection lines. You can execute the code in the Python window of ArcMap after changing input data names and maxdistance (the max distance between ends for a connection line to be created)
#Change these four lines to match your data (whats on the right side of the equal to signs)
arcpy.env.workspace = r'C:\Users\bera\Documents\ArcGIS\Projects\MyProject\MyProject.gdb'
bike = 'Bike'
walking = 'Walking'
connect = 'Connect' #Name of the new feature class to be created
maxdistance = 100
#List all start and endpoints as pointgeometries
bike_ends = [arcpy.PointGeometry(j) for k in [[i.firstPoint, i.lastPoint] for i in arcpy.da.SearchCursor(bike,'SHAPE@')] for j in k]
walking_ends = [arcpy.PointGeometry(j) for k in [[i.firstPoint, i.lastPoint] for i in arcpy.da.SearchCursor(walking,'SHAPE@')] for j in k]
#Find and list the pairs within maxdistance
ends_to_connect = 
for b in bike_ends:
for w in walking_ends:
if b.distanceTo(w) <= maxdistance:
#Create polylines from point pairs
polylines = [arcpy.Polyline(arcpy.Array(pair)) for pair in ends_to_connect]