Katherine Needs a Drink is an opportunity given by Katherine. It is related to Frank's opportunity Look at This Photograph!, Katherine's opportunity Katherine's Dilemma, and Stephanie's Apply Pressure. Its conclusion (with success or failure) unlocks Katherine's Game Jam opportunity to bring the Game Grumps to the party.
Speak with Katherine. Pander to her erudite superiority by commenting on pedantic use of the word "need". Followup by commenting on her attention to her phone, and ask who would ever be a dick to her. Then, either follow up by offering to get her a drink, or comment that she is still using her phone; both options result in Katherine accepting (or demanding) your drink (Opportunity Katherine Needs a Drink discovered).
In short, you need to get Katherine some booze.
You can then ask Katherine what kind of drinks she likes. She states that she wants rum. Unfortunately, the only rum in the house is in the liquor cabinet in the Dining Room, guarded by Frank.
Before speaking with Frank, right-click on the Rum to take it in your hand. Frank will warn you to put it down. Put it down out of his sight (in front of the stove in the Kitchen works), then pick it up out of sight of both Leah and Frank.
Alternately, talk to Frank, then talk to Patrick. Warn Patrick about Frank, then ask Patrick how he hides his wine. Narc on Patrick to Frank, then get the Rum from the Liquor Cabinet while he's away from his post.
Give Katherine the Rum (Opportunity Katherine Needs a Drink complete).
Ask Frank whether he's just gonna guard the alcohol, then ask his opinion of Katherine.
Go to Katherine and tell her that Frank's guarding the alcohol. She will express frustration, but offer to help you work around the problem if possible. Optionally, inform her that Frank likely has a crush on her to see her reaction.
Go to Frank and offer to talk to Katherine on his behalf. He will decline, but ask for topless photos of Katherine (Opportunity Look at This Photograph! discovered).
Get topless photos of Katherine
Tell Katherine of Frank's offer, and she will decline (Opportunity Katherine's Dilemma discovered).
Go to the garage and collect the Cell Phone Jammer, then go to Madison's bedroom and collect the Camera, then to the Study Room and collect the SD Card. Insert the SD Card into the Camera, then move to Katherine and Turn On the Cell Phone Jammer. Keep it on and follow Katherine until she walks to a place in the house where you can be alone with a closed door (typically either the Garage or the Art Room).
With the door closed, ask about taking the topless pic, and Katherine will agree (Opportunity Katherine's Dilemma complete). Talk with her again, and ask her to take her top off. Talk to her again and ask her to go truly topless. Take the photo with the Camera.
Give the topless photos to Frank
After you have collected the topless pic(s), you can ask her to take a break or simply leave the room. Return to Frank and give him the Camera (Opportunities Katherine's Dilemma and Look at This Photograph! complete). He will allow you to take one item from the cabinet. Take the Rum.
Completing Katherine's opportunity
Return to Katherine and, out of sight of both Frank and Leah, give her the Rum (Opportunity Katherine Needs a Drink complete).
If you are trying to complete Stephanie's quests, after you acquire the rum, find the Laxative in Madison's closet, then use the Laxative with the Rum to get Tampered Rum. Give the Tampered Rum to Katherine (Opportunities Katherine Needs a Drink and Katherine's Dilemma missed, opportunity Apply Pressure discovered).
Talk with Katherine again and take a swig of the rum with her to discover Game Jam. Talk with her and ask about a 100% chance of getting into her pants to discover Express Yourself. If you gave her Tampered Rum, simply wait a couple of minutes to complete Apply Pressure and move on with Stephanie.