If a travel agent or someone in an allied industry needs a copy of your passport, you can always send an image with redacted information. This is an example from the Home Office site...
As can be seen, information which is exempt is blotted out (blue rectangles). It shows, yes, you are a British citizen; yes, you have a current passport, and yes, it was issued in the UK. That's all that should be required. Secondary abuse by personnel handling your image at the travel agency is eliminated.
In some jurisdictions, redaction is a legal requirement. Consider this extract from Texas, US which lists the information that should be redacted...
Unless the Court orders otherwise, an e-filed document must not
contain a social security number; a birth date; a home address; the
name of any person who was a minor when underlying suit was filed; a
driver’s license number, passport number, tax identification number,
or similar government-issued personal identification number; or a bank
account number, credit card number, or other financial account number.
The e-filer must redact all of this information in accordance with the
redaction guidelines posted by the Court’s Clerk on the Court’s
website; however, the e-filed document may contain a reference to this
information as long as the reference does not include any part of the
actual information (e.g., “passport number”). For good cause, the
Court may order redaction of additional information
In the case of Mass, US, they indicate that the last four digits of the passport number be excluded from redaction...
in the case of a social security number, taxpayer identification
number, credit card or other financial account number, driver’s
license number, state-issued identification card number, or passport
number, only the last four digits; and
The rules for redaction are different depending upon jurisdiction. In the UK, bank statements submitted for visa evidence can have the account numbers redacted.
If a travel agent objects to redaction, especially if the courts in their jurisdiction allow it (i.e., they are over and above their own government), then you should consider if the staff do not understand the requirements, or finally that the agent should not be trusted. In all events, if you really NEED to send exempt information, you can send it via separate cover.
Airlines and other entities that have a LEGITIMATE need for your passport number and DOB invariably use secure channels anyway.
NOTE: for purposes of this answer, I included the right hand side of the portrait page only. There is nothing on the left hand side that needs redaction unless the passport holder is a minor.