If the engine runs without having the battery connected, the alternator is probably OK.
I would check for a blown fuse (or fuses) and if none are bad, the relays that do most of the switching of different accessories would be the next place to look.
Be advised, if the battery is disconnected with the engine off, you may lose access to your stereo and possibly the ability to lock and unlock your car with a remote.
This is a common problem and part of the passive security in most cars that is supposed to prevent theft since the codes can disappear with power removed.
You can prevent this from happening with a simple and low-cost device that uses a 9V battery and plugs into the cigarette lighter or 'power jack.' The battery keeps those items 'alive' that are protected as described.
I recommend to anyone owning any vehicle to spend ~$25 US on a Haynes service manual (if one exists) which covers mechanical and electrical service.
We have one for each of three vehicles.
For yours:
VW New Beetle manual
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