How do I address a small crack on the bezel of my MacBook Air?

Be aware that any damage to the display on Mac laptops made since about 2013 requires the replacement of the entire sealed display assembly. A single component of the display cannot be separately replaced.

W are not Apple so can cannot quote costs, but others posting similar issues here have reported costs of US$500 and up (parts + labor) for a new assembly. Therefore, depending on the year model of your Macbook Air (which we do not know), you need to ask yourself:

1 — Is this computer new enough (runs latest macOS version) to repair?

2 — Is it old enough to a justify new computer?

3 — Does Apple still provide service for my model?

This Apple support article shows models they consider vintage or obsolete and no longer service:

Obtaining service for your Apple product after an expired warranty - Apple Support

Take the computer to your local Apple Authorized Service Provider to have it inspected.

You can try cutting some black electrical tape to put over the crack if you have about an inch on either side. Otherwise I would very carefully put some glue along the crack to hold it together. After all the damage is already done and it will be extremely expensive to have the entire Display Assembly replaced.

Technically it is possible to replace just the Bezel, but it is very tricky & risky even for an experienced tech. I've done it multiple times, but I don't know of anyone else who has done so.

If that crack is due to an inflating battery, you don't want to mess with the computer, bring it in for service. Older batteries can often get a chemical imbalance and need to be replaced. Power down the machine if it appears the battery is what broke the bezel and take it in for service.