Nowadays, it is possible to get your KYC done online in the following ways:

Aadhar-based e-KYC:

You need to provide your Aadhar number, the registered mobile number (linked with the aadhar card) and the OTP (one time password) which is sent to this registered mobile number. You also need to upload a scanned self-attested copy of aadhar card. Which this method is completely electronic, it comes with certain restrictions. It is available only for investments where the mode of holding is single. Further, a person KYCed through this method and can invest up to a maximum of Rs 50,000 in one fund house in a financial year. For more details and to get your KYC done, visit this link 

PAN-based e-KYC:

Some fund houses also offer the facility of getting your KYC done electronically based on your PAN card. You need to fill the relevant form online, take a print out and upload its scanned copy with your photograph and signature, along with other relevant documents. Subsequently, your in-person verification is done through a webcam or your smartphone, without you having to go anywhere.

The main advantage of this method is that the restrictions, as applicable on Aadhar-based KYC do not apply here. Birla Sun Life Mutual Fund is one of the few AMCs that offer both these methods of KYC on its website.