Here's one way to do it. This example is based on a default Blazor Server project, with the Index.razor, Counter.razor, and FetchData.razor components/pages.

Add a scripts folder to your wwwroot, if you don't already have one, add a new JavaScript file called 'body-class-changer.js" and insert the following code

window.bodyClassMethods = {
    setBodyClass: function (bodyClass) {
        document.body.classList.value = bodyClass;
    }
}

Add a script tag to the bottom of your _Host.cshtml file, just above the closing body tag

<script src="~/scripts/body-class-changer.js"></script>

Add a class file to your Shared folder called "ClassSelector.cs" and adjust the class with the following code

public class ClassSelector
{
    public static string GetClass(string currentPage)
    {
        switch (currentPage)
        {
            case "": 
                return "home-class";
            case "counter":
                return "counter-class";
            case "fetchdata":
                return "fetchdata-class";
        }
        return string.Empty;
    }
}

Add a new Razor Component to your Shared folder called "ClassChanger.razor" with the following code

@inject NavigationManager navigationManager 
@inject IJSRuntime jsRuntime
@implements IDisposable

@code {
    string bodyClass { get; set; }

    protected override void OnInitialized()
    {
        navigationManager.LocationChanged += OnLocationChangedAsync;
    }

    protected override async Task OnAfterRenderAsync(bool firstRender)
    {
        if (firstRender)
        {
            bodyClass = Shared.ClassSelector.GetClass(navigationManager.ToBaseRelativePath(navigationManager.Uri));
            await jsRuntime.InvokeVoidAsync("bodyClassMethods.setBodyClass", bodyClass);
        }
    }

    public void Dispose()
    {
        navigationManager.LocationChanged -= OnLocationChangedAsync;
    }

    private async void OnLocationChangedAsync(object sender, LocationChangedEventArgs args)
    {
        bodyClass = Shared.ClassSelector.GetClass(navigationManager.ToBaseRelativePath(navigationManager.Uri));
        await jsRuntime.InvokeVoidAsync("bodyClassMethods.setBodyClass", bodyClass);
        StateHasChanged();
    }
}

Then add this Razor Component to the bottom of your MainLayout.razor file

<ClassChanger></ClassChanger>