Deleting previous version folder of android studio

No, you're good to go removing the prior version's directory which in this case is the .AndroidStudio1.3 directory.

On a side note, make sure you're importing settings from 1.3 when upgrading to 1.4. That will spare the Android Studio from re-downloading all the existing project dependencies you worked with before.

It depends.
Android Studio stores inside these folders the configuration files of the specified version.
You can remove these folders, if you are not using them anymore.

When you install a new version, Android Studio asks you if you want to import the previous settings.

In any case:

.AndroidStudio1.2          Used by Android Studio 1.2
.AndroidStudio1.3          Used by Android Studio 1.3
.AndroidStudio1.4          Used by Android Studio 1.4
.AndroidStudioBeta         Used by Android Studio Beta
.AndroidStudioPreview      Used by Android Studio Preview
.AndroidStudioPreview1.3   Used by Android Studio Preview 1.3

TL;DR: No. It will not cause any problems.

Hidden folders in your home directory should contain only application settings. If you remove one, you will simply revert AndroidStudio settings to default.

In this case, you have a separate directory for two versions of AndroidStudio. If you remove .AndroidStudio1.3, you will simply reset settings in 1.3, leaving 1.4 unaffected.

However, you may consider copying settings from 1.3 to 1.4 before removing them.