Setting width of layers panel in QGIS?

Using a bit of Python, we could create a startup.py script which sets the width of the layers panel by using the following:

from qgis.utils import iface
from PyQt4.QtGui import QDockWidget

for x in iface.mainWindow().findChildren(QDockWidget): 
    if x.objectName() == 'Layers':
        x.setFixedWidth(250)
        x.setMinimumWidth(100)
        x.setMaximumWidth(1000)

Save the script in your /.qgis2/python/ directory. Now whenever you load QGIS, the width of the layers panel will be set accordingly.

For reference: there is Bug report #15479 Layers panel does not remember size which is related to your issue.