I use barley tea bags infusing in cold water. Is there any hygienic problem? Is it better to use boiling water?

Whether using cold water or boiling water, proliferation of bacteria is not so different.  What is important is hygienic control of the container.

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There was a complaint sent to a health center in Tokyo, saying, “I drank purchased tea directly from the PET bottle by putting my lips at the bottle mouth, and then left the remainder in a refrigerator for several days.  And this caused mold to grow.”
Even if the food, such as tea, has no problem, it will be contaminated if the container used to keep it is contaminated.  In the same way, if hands are dirty when tea is prepared, the tea and the container will also be contaminated.
How long homemade drinks and leftover drinks can be kept is not defined.  The “best before” date indicated on the PET bottle does not guarantee the quality after opening.  Appropriately control the temperatures of the drinks and quickly consume them.
When you prepare a drink at home, whether cold water or boiling water is used, carefully wash your hands and the container and handle the teabag and the water in a hygienic manner.  In addition, do not directly drink from the storage container but use a glass, etc., and consume it up in a short period.