With the fondue
What is more natural than to drink the same (or better) white wine than the one used to make the fondue (see the wine section).
If you do not drink alcohol, try a black tea (avoid if possible to put sugar in it).
Avoid at all costs, cold drinks in large quantities: beer, water or other. These cold beverages will make digestion difficult (possibly very difficult) solidifying the cheese in the stomach (so it seems).
The wine does not have this drawback because it is drunk in small sips.
White wine is served cool (6 to 10°C - 43 to 50°F)