So you want to change out the whole enclosure (reflector, bulbs, and all) to a 2-bulb system?
It's simple; your existing headlight draws 60 watts (/ 14 volts = about 4 amps) on high beam, the new system with two bulbs will draw 120 watts / 14 volts = about 8 amps.
The starting motor is a completely unrelated circuit so there is no way this could cause any direct damage ... but if your proposed new headlight system draws out electricity faster than your alternator can supply (worst case is idling in traffic jam) it will slowly discharge the battery and thus you might discharge the battery to the point where the engine may not start because there is not enough charge left in the battery ... maybe that was behind your misinterpretation of what someone told you ...
I have no idea what the capacity of your alternator is, nor the current-carrying capacity of your headlight switch, nor the wiring from the fuse box to the switch and then to the headlights. Your best source of that information is someone who has done this before.
For what it's worth, OEM headlights with a high/low beam that are similar to the normal 60/55 watt high/low H4 are available that are 35/35 watt; for example the OEM headlight bulbs in Yamaha FZR400/600, FZ600, etc are like this. They're about as effective as holding up a candle in front of the bike. And those bikes do not have enough reserve capacity in the electrical system to drive two 60/55 watt bulbs through the stock wiring and switch. With the engine idling, headlight and instrument cluster indicator brightness varies in opposition to the flash cycle of the turn signals even with the stock headlights ...