Airport transit visa Frankfurt

Timatic is the database which the airline (Lufthansa?) uses to decide whether to let you board the flight, and it says the following:

National India (IN)             /Transit Germany (DE)
Destination USA (US)    

[Visa Information - Transit] Germany (DE)

[Information For Normal Passports]
TWOV (Transit Without Visa):
Visa required, except for Nationals of India transiting
through Frankfurt (FRA) or Munich (MUC), arriving from a
non-Schengen Member State with a confirmed onward ticket for a
flight within 24 hours to a third country which is not a
Schengen Member State. They must:
- have a visa issued by Bulgaria, Canada, Croatia, Cyprus,
  Ireland (Rep.), Japan, Romania, USA or United Kingdom, and
- stay in the international transit area of the airport, and
- have documents required for their next destination.

So you're fine if you have a valid visa issued by the United States, which your F-1 visa is.

It would surprise me if the airline agent knew anything about your U.S. immigration status, but let's still distinguish two cases:

• If they know about your H-1B status and request to see the H-1B visa, answer truthfully that you haven't gotten it yet, but you have an old visa (F-1) with you that is still valid.
• If they don't know about your H-1B status (and this should be the case), then don't mention it at all and just present your unexpired, valid F-1 visa, which permits you to transit Frankfurt visa-free.