You can use a script like this. I used row and col, so you can drag the formula in case you need it:

You can use it like: =getBorderInfo(ROW(B1),COLUMN(B1))

function getBorderInfo(row, col) {
  var sheet = SpreadsheetApp.getActiveSpreadsheet().getActiveSheet();
  
  // Get the current cell's range
  var range = sheet.getRange(row, col);
  var borders = [0, 0, 0, 0]; // Initial value for [left, top, right, bottom]
  
  // Check the current cell's borders
  var border = range.getBorder();
  if (border) {
    if (border.getLeft().getBorderStyle() !== null) borders[0] = 1;
    if (border.getTop().getBorderStyle() !== null) borders[1] = 1;
    if (border.getRight().getBorderStyle() !== null) borders[2] = 1;
    if (border.getBottom().getBorderStyle() !== null) borders[3] = 1;
  }
  
  // Check left adjacent cell for its right border
  if (col > 1) {
    var leftCell = sheet.getRange(row, col - 1);
    var leftBorder = leftCell.getBorder();
    if (leftBorder && leftBorder.getRight().getBorderStyle() !== null) borders[0] = 1;
  }
  
  // Check top adjacent cell for its bottom border
  if (row > 1) {
    var topCell = sheet.getRange(row - 1, col);
    var topBorder = topCell.getBorder();
    if (topBorder && topBorder.getBottom().getBorderStyle() !== null) borders[1] = 1;
  }
  
  // Check right adjacent cell for its left border
  if (col < sheet.getMaxColumns()) {
    var rightCell = sheet.getRange(row, col + 1);
    var rightBorder = rightCell.getBorder();
    if (rightBorder && rightBorder.getLeft().getBorderStyle() !== null) borders[2] = 1;
  }
  
  // Check bottom adjacent cell for its top border
  if (row < sheet.getMaxRows()) {
    var bottomCell = sheet.getRange(row + 1, col);
    var bottomBorder = bottomCell.getBorder();
    if (bottomBorder && bottomBorder.getTop().getBorderStyle() !== null) borders[3] = 1;
  }
  
  
  return borders.join("");
}

Update Workaround:

Script-based formulas only update when the value of a cell they use is updated. What you can do (depending on how many times you’re using it all over the spreadsheet) is to create an “updater cell” that is used only for that. Let’s say it’s Z1 – Then you use =getBorderInfo(ROW(B2),COLUMN(B2),$Z$1) as a third argument; and an onEdit(e) function that updates Z1 with each modification of the spreadsheet:

function onEdit(e){
  SpreadsheetApp.getActive().getRange("Z1").setValue(new Date())
}

Then every time a border or anything is modified, it would update Z1 which would eventually update every script-based formulas. You can think of ways to have an “updater column” and modify onEdit to update the row of the cell, or something like that