Ok, so napkin math time:
Assume you need 3 people to drive out to apprehend illegal crossings for officer safety. You also then need 3 others to watch while others are out apprehending and then be able to respond for other illegal crossings.
Assume 3 minutes to cross over the barriers per person.
Assume it takes you no time to see the crossing happen, you can tell the second a ladder is put up.
Assume the roads are dirt, but in good and dry conditions. So you can travel about 40 mph on them on average, including the time to accelerate from 0mph at the remote watch tower station and then back down to 0 to apprehend people. 40mph is 2/3 miles per minute, btw.
So, the minimum distance at 3 minutes to cross would then be 2 miles. Meaning you'd need about 6 guys every 2 miles. So about 12,000 people for the 2,000 mile border.
Now you can play with the numbers, and talk about capturing people that scatter into the interior, but you're still at about 1 officer per 1 mile (ish), not 1 officer per 10 miles or 100 miles of border, with significant downtime per officer per day.