When does the 'Date' occur at Golden Saucer?

Go to the Temple of Ancients and you will find that you need a Keystone, which belongs to the eccentric owner of a well-known theme park.

Here's a good guide that goes over how to take Barrett on the date you are trying to get to: http://www.princessartemis.com/documents/ff7date.txt

Starting affection:
  Aeris:  50
  Tifa:   30
  Yuffie: 10
  Barret:  0

Barret:
1. Buy a flower from Aeris when you see her in Sector 8.
2. Talk to Jessie twice on the train home from Sector 8, and tell her
    you're "Looking forward to it".
3. Give the flower to Marlene when you get the option.
4. Don't talk to Tifa while she's behind the bar.  If you do, decline her
    offer for a drink.
5. Say "...Sorry" when she asks you if you've forgotten your promise.
6. Say "Barret's snoring kept me up..." when she asks you how you slept the
    next morning.
7. Don't get caught by the Security Lockdown; get to the end of the train.
8. After defeating Airbuster, before Cloud falls, tell Barret to
    "Be strong".
9. When Aeris is fleeing the guards, climb up to the top level and tell her
    to Wait each time.  Then, push the *wrong* barrel (so that she has to
    fight the guard anyway).  Do this three times.
10. When Aeris asks if Tifa is your girlfriend, say "Yeah, that's right."
11.  When you get to the playground in Sector 6, tell Aeris that you'll
    "Take her home."
12. Get *all* the best items so that Cloud is chosen for the Don's date
    (not Tifa or Aeris).  Tell the Don that there's someone else, and when
    he asks who, tell him "Yes, his name's Barret..."
13. When everyone lands in the sewer, talk to Aeris first.
14. When Marlene asks you how you feel about Aeris, tell her
    "I don't know".
15. When you first meet Red XIII, tell Barret to take care of Aeris (keep
    Tifa in your party for *this* battle).
16. After the battle with HO512, split up into a team of Cloud, Barret and
    Red XIII.
17. When Tifa asks if you can break out, tell her "(Kinda hard)"
18. Think about Barret first, Red XIII second, and Aeris last while you're
    in the cell.
19. When you finally escape Midgar and have to choose your team, pick a
    team with Cloud, Barret and Red XIII.
20. In Kalm, after the flashback, tell Barret to "Wait a sec", then
    "Beautiful, just beautiful!"
21. Make a party immediately afterwards with Barret, Cloud and Red XIII.
    In a house in Kalm, there's a woman who asks you about whether Mako
    energy is convenient.  Tell her "You're full of it".  Talk to the girl
    upstairs, and when she asks you if the old life is better, say 
    "Yeah, maybe".
22. In Fort Condor, agree twice to the old guy about his plight.
23. You don't have to meet Yuffie; in fact, it's probably better if you
    don't.  But you can if you want to.  Just make sure you get her the
    first time, and don't make her run away.
24. On the Cargo Ship, don't give Yuffie a tranquilizer.
25. On the Cargo Ship, tell Aeris "I dunno..." when she asks about the
    Airship.
26. On the Cargo Ship, when Tifa asks you about war, say "I don't know".
27. At the Gold Saucer, when you split up, talk to Red XIII first, and take
    him with you (do not choose Aeris, Tifa or Yuffie).
28. When you see Barret again in the Corel Prison, and you're about to go
    after Dyne, you will be prompted to make a party.  Choose Cloud, Barret,
    and either Red XIII or Cait Sith.  *DO NOT CHOOSE AERIS*!  She gets a
    heck of a boost if she gets chosen, and will mess up everything you've
    worked for.
29. Before entering Gongaga Town, make your party into Cloud, Aeris and
    Tifa.  Then, when you meet Zack's parents, and Aeris and Tifa leave,
    ignore *both* of them and leave town.
30. Do not go anywhere near Wutai if you have Yuffie.  Instead, go straight
    to the Gold Saucer for the date sequence.

Ending Affection Scores
  Barret: 47
  Aeris:  37
  Tifa:   35
  Yuffie: 14

Since it's not clear enough to continue on after that point to trigger the date scene at the gold saucer, starting from after you get the tiny bronco.

1. Take Tiny Bronco to west of Gongaga to a tiny house with an arms dealer.
2. Talk to Arms Dealer, he tells you to go to Gold Saucer
3. Go to Gold Saucer, talk to Dio in battle Square.

Just keep following the story at gold saucer, and the date sequence happens after you go to your room in the Ghost Square. The Arms Dealer is supposed to be optional, but if he talks about the keystone, you've already triggered the event at the Gold Saucer.