That's a great question. I've worked with many companies which, because they did not ask that question, experience a massive nose dive in their culture when they laid off people.
Off the top of my head, some ways to minimize the impact of downsizing on culture, are ....
1. Make sure the CEO and execs are role modeling to the nth degree the values and behavior norms implementing/communicating the layoff.
For example, if respect is a core value and a behavior norm is acknowledging people for their contributions, then applying that value/norm would first treat laid off people with respect and dignity and acknowledge publicly to whole company their contributions in the company. I know one CEO that went as far as making sure every downsized employee found a job in another company by making connections/recommendations to those in their network.
2. Transparency to the nth degree is huge. Letting the company know well in advance this change is happening, why it is happening, how decisions will be made and steps company will take specifically to move through transition.
To not have extreme transparency breeds mistrust, fear, fragmentation and others leaving as well.
3. Open honest dialogue where those remaining can voice their feelings and concerns.
Downsizing is a real jolt to an organization and lots of emotions come up as it is a form of loss. There needs to be time for grieving, voicing any apprehensions and asking questions.
Leaders need to be aware of how employees respond to the change. Do some adapt to the change quickly? are there resistors? how will you deal with resistors? will fear drive a wedge between teams/functions?
4. Plan for and communicate other related changes.
Those employees left behind may get transferred to new role. Or they may take on other's responsibilities, aside from their own. Or they may report to a different boss. Etc.
This need to be well laid out and communicated in advance as to minimize confusion and chaos.
Also have the goals, markets, policies, etc of the company changed? is that part of the reason for the downsize? All this needs to be communicated multiple times in multiple ways.
5. Take a pulse on the organizational climate 1 month, 3 months and 6 months out.
You want to keep your eye on if or to what extent the culture has eroded for things that were not obvious right away.
Once the organization gets through the most difficult part of the transition period, it's time to rebuild trust and re-inspire. Employees look to leaders during those times for positive inspiration. They want to be re-energized and believe that leaders have their back.
Those are the first things that come to mind to your question. Appreciate the discussion.