As most El Capitan users already found out: The Finder sidebar is messed up and doesn't remembers it's width after changing it. User aatrcoutinho already posted, on the past two Finder threads, a workaround that includes.. a bunch of code. Although it's the most light-weight option of that- and this one, I do prefer an easier approach. And if you ask me: A better functioning way.
My approach is to use 'Keyboard Maestro'. Just download the trial (fully functional, works until El Capitan is released). It might seem as much work, but it's quite easy and it pays off. You could do this within 5 minutes. Remember to set it's preferences to 'Launch engine at login', else the macros won't load once you reboot.
To get it working you need to create three macros.
1. Set width when Finder starts
A. Select the 'Global macro group'
B. Press the '+' below to create a new macro
C. Set name to whatever you like (i.e. 'Set width when Finder activates')
D. Press 'new trigger' and choose 'This application' and choose 'Finder' and below choose 'Activates'.
E. Press 'new trigger' and choose 'This application' and choose 'Finder' and below choose 'Launches'
F. Press 'New action' and search for 'Execute an AppleScript'. Double click it.
G. Input in the box 'tell application "Finder" to set the sidebar width of every Finder window to 160'
Optional: Change the default width by changing the number ('160')
H. Press 'edit' at the bottom of the window to save your macro.
Now your Keyboard Maestro should look like this
2. Set width when opening a secondary Finder window
A. At the 'groups' column, click on the '+' to create a new Macro group. Let's call it "Finder macro's"
B. Set 'Available in all applications' to 'Available in these applications' and add 'Finder' to it. Press the 'edit' button below to save the group.
C. Select the new group and in the 'macros' column press the '+' to create a new macro. Call it whatever you like (i.e. 'New window')
D. Press 'New trigger' and choose 'This hotkey' and put in 'CMD + N'
E. Press 'New action' and search for 'Select or show menu' and double click it.
F. Set the input of 'menu title' to 'File' and set the input of 'menu item' to 'New Finder-window'
G. Press 'New action' and search for 'Execute an AppleScript'. Double click it.
H. Input in the box 'tell application "Finder" to set the sidebar width of every Finder window to 160'
Optional: Change the default width by changing the number ('160')
I. Press 'edit' at the bottom of the window to save your macro.
Now this macro will look like this:
3. Set width on new tab
A. Select the just created 'New window' macro and copy paste it (CMD+C, CMD+V) and press 'edit' to edit it.
B. Change it's name to a new name (i.e. 'New tab')
C. Change the hotkey input from 'CMD+N' to 'CMD+T'
D. Change the action from 'Menu item', 'New Finder-window' to 'New tab'
E. Press 'edit' to save this last macro.
This macro will look like:
Now enjoy using Finder with a good sidebar width again.
Feel free to add suggestions
Note: If you are annoyed by the new tab switcher, just like me, head over, in Keyboard Maestro, to the 'Switcher group' group and select all macro's and either remove them (CMD+delete) or press the check sign below.
My approach is to use 'Keyboard Maestro'. Just download the trial (fully functional, works until El Capitan is released). It might seem as much work, but it's quite easy and it pays off. You could do this within 5 minutes. Remember to set it's preferences to 'Launch engine at login', else the macros won't load once you reboot.
To get it working you need to create three macros.
1. Set width when Finder starts
A. Select the 'Global macro group'
B. Press the '+' below to create a new macro
C. Set name to whatever you like (i.e. 'Set width when Finder activates')
D. Press 'new trigger' and choose 'This application' and choose 'Finder' and below choose 'Activates'.
E. Press 'new trigger' and choose 'This application' and choose 'Finder' and below choose 'Launches'
F. Press 'New action' and search for 'Execute an AppleScript'. Double click it.
G. Input in the box 'tell application "Finder" to set the sidebar width of every Finder window to 160'
Optional: Change the default width by changing the number ('160')
H. Press 'edit' at the bottom of the window to save your macro.
Now your Keyboard Maestro should look like this

2. Set width when opening a secondary Finder window
A. At the 'groups' column, click on the '+' to create a new Macro group. Let's call it "Finder macro's"
B. Set 'Available in all applications' to 'Available in these applications' and add 'Finder' to it. Press the 'edit' button below to save the group.
C. Select the new group and in the 'macros' column press the '+' to create a new macro. Call it whatever you like (i.e. 'New window')
D. Press 'New trigger' and choose 'This hotkey' and put in 'CMD + N'
E. Press 'New action' and search for 'Select or show menu' and double click it.
F. Set the input of 'menu title' to 'File' and set the input of 'menu item' to 'New Finder-window'
G. Press 'New action' and search for 'Execute an AppleScript'. Double click it.
H. Input in the box 'tell application "Finder" to set the sidebar width of every Finder window to 160'
Optional: Change the default width by changing the number ('160')
I. Press 'edit' at the bottom of the window to save your macro.
Now this macro will look like this:

3. Set width on new tab
A. Select the just created 'New window' macro and copy paste it (CMD+C, CMD+V) and press 'edit' to edit it.
B. Change it's name to a new name (i.e. 'New tab')
C. Change the hotkey input from 'CMD+N' to 'CMD+T'
D. Change the action from 'Menu item', 'New Finder-window' to 'New tab'
E. Press 'edit' to save this last macro.
This macro will look like:

Now enjoy using Finder with a good sidebar width again.
Feel free to add suggestions
Note: If you are annoyed by the new tab switcher, just like me, head over, in Keyboard Maestro, to the 'Switcher group' group and select all macro's and either remove them (CMD+delete) or press the check sign below.