I used to attend regularly, starting back in the 1960's. The draft was held during the week at a hotel in midtown Manhattan. I used to take a long lunch hour and go back for a couple of hours after work. The draft gurus of that era were Palmer Hughes, The Jerry Jones Drugstore List, Joel Buchsbaum and Mel Kiper.
The League attitude toward fans attending was that we were trying to sneak in, but that gradually changed and the NFL became more lenient, reserving limited space in the balcony for draftniks on a first come/first served basis. The draft was not televised except for a minute or two on a highlight show.
They then moved the draft to Saturdays and Sundays (& at 7 am, there would be a long line of early attendees). By then, there was a group of us regulars who'd network throughout the year and sit together at the draft. At the time, Phil Hepler and Bill Werndl of Ourlads joined Kiper along with David Te Thomas who used to subcontract for the NFL to provide college player stats for the 32 teams.
By mid-1980's, I had left my corporate job in NYC and hooked up as Promo Mgr for a NJ daily. With that gig came press credentials (along with desk space and buffet food).
Right about then, the draft was becoming a big deal, with the League marketing it like a spectator sport. ESPN was now covering the first day of the draft around the clock. I got lazy - finding it easier to follow the draft on cable tv or my computer rather than schlep my way into NYC.
Of course, now the draft is a very big deal - with more coverage than Beyonce's cleavage. It's a 3 day extravaganza which will move to different cities like the Super Bowl. It's worth attending once, but the crowds may now be ferocious, and you really can follow it better from home nowadays.
But those were the days. My favorite memory was going off for a run up the east river drive in 1981 knowing that we had just drafted EJ Junior and returning to the Vanderbilt "Y" (on 41st St) to discover that we drafted Neal Lomax in the second round. (Note - We also grabbed Stump Mitchell in the 9th round).
Another memory had to do with the year QB Dan Pasterini was drafted by Houston. The Runyonesque corporate head of marketing at the corporation where I worked called me into his office in mid-afternoon: "How's the draft going?" ("Fine"). "Where did Pastrami get drafted?"...
If there was no such thing as the NFL Draft, we'd have to make it up.