How can I tell Swashbuckle that the body content is required?

To achieve this you have to do a couple of things.

First you have to tell Swagger there's a parameter in the body that contains binary data. Next you have to tell Swagger that the end point consumes binary data (e.g. application/octet-stream).

Swashbuckle does not support this out of the box. But you can create custom filters to extend the functionality of Swashbuckle. What I usually do is create a custom attribute to decorate a method and then create a custom filter to act upon that attribute.

In your case this would do the trick:

The custom attribute

public class BinaryPayloadAttribute : Attribute
{
    public BinaryPayloadAttribute()
    {
        ParameterName = "payload";
        Required = true;
        MediaType = "application/octet-stream";
        Format = "binary";
    }

    public string Format { get; set; }

    public string MediaType { get; set; }

    public bool Required { get; set; }

    public string ParameterName { get; set; }
}

The custom filter

public class BinaryPayloadFilter : IOperationFilter
{
    public void Apply(Operation operation, SchemaRegistry schemaRegistry, ApiDescription apiDescription)
    {
        var attribute = apiDescription.GetControllerAndActionAttributes<BinaryPayloadAttribute>().FirstOrDefault();
        if (attribute == null)
        {
            return;
        }

        operation.consumes.Clear();
        operation.consumes.Add(attribute.MediaType);

        operation.parameters.Add(new Parameter
        {
            name = attribute.ParameterName,
            @in = "body", 
            required = attribute.Required,
            type = "string", 
            format = attribute.Format
        });
    }
}

Add the filter to the Swashbuckle configuration

GlobalConfiguration.Configuration 
    .EnableSwagger(c => 
        {
            // other configuration setting removed for brevity
            c.OperationFilter<BinaryPayloadFilter>();
        });

Apply the attribute to your method

[HttpPost]
[BinaryPayload]
[Route("Store/{projectId}")]
public async Task Store(string projectId)
{
    ...
}

In Swagger UI you then get:

In Swashbuckle 4.0 the syntax has changed a little:

public class BinaryPayloadAttribute : Attribute
{
    public BinaryPayloadAttribute()
    {
        ParameterName = "payload";
        Required = true;
        MediaType = "application/octet-stream";
        Format = "binary";
    }

    public string Format { get; set; }
    public string MediaType { get; set; }
    public bool Required { get; set; }
    public string ParameterName { get; set; }
}

public class BinaryPayloadFilter : IOperationFilter
{
    public void Apply(Operation operation, OperationFilterContext context)
    {
        var attribute = context.MethodInfo.DeclaringType.GetCustomAttributes(true)
            .Union(context.MethodInfo.GetCustomAttributes(true))
            .OfType<BinaryPayloadAttribute>().FirstOrDefault();
        if (attribute == null)
        {
            return;
        }

        operation.Consumes.Clear();
        operation.Consumes.Add(attribute.MediaType);

        operation.Parameters.Add(new BodyParameter
        {
            Name = attribute.ParameterName,
            @In = "body",
            Required = attribute.Required,                                
            Schema = new Schema
            {
                Type = "string",
                Format = attribute.Format
            }
        });
    }
}

And:

services.AddSwaggerGen(c =>
{
    c.OperationFilter<BinaryPayloadFilter>();
});

Yet another update. Here's the solution I ended up with using ASP.NET Core 3.1 and Swashbuckle.AspNetCore.Swagger 5.0.0:

public class BinaryContentAttribute : Attribute
{
}

public class BinaryContentFilter : IOperationFilter
{
    /// <summary>
    /// Configures operations decorated with the <see cref="BinaryContentAttribute" />.
    /// </summary>
    /// <param name="operation">The operation.</param>
    /// <param name="context">The context.</param>
    public void Apply(OpenApiOperation operation, OperationFilterContext context)
    {
        var attribute = context.MethodInfo.GetCustomAttributes(typeof(BinaryContentAttribute), false).FirstOrDefault();
        if (attribute == null)
        {
            return;
        }

        operation.RequestBody = new OpenApiRequestBody() { Required = true };
        operation.RequestBody.Content.Add("application/octet-stream", new OpenApiMediaType()
        {
            Schema = new OpenApiSchema()
            {
                Type = "string",
                Format = "binary",
            },
        });
    }
}

ConfigureServices in Startup.cs:

services.AddSwaggerGen(o =>
{
    o.OperationFilter<BinaryContentFilter>();
});