You could apply the condition using ternary operator for label and use [disabled] property to enable/disable the button. You could take advantage of 0 being falsy in JS.
Try the following
<td *ngFor="let application of applications; let index=index">
  <p> 
    <button (click)="showInfoWindow(application)" [disabled]="!!index" class="btn btn-sm btn-primary">
      {{ !!index ? 'INFO' : 'info' }}
    </button>
  </p>
</td>
Update: Using *ngIf
<td *ngFor="let application of applications;let index=index">
  <p> 
    <button *ngIf="index; else enabled" (click)="showInfoWindow(application)" disabled class="btn btn-sm btn-primary">
      INFO
    </button>
    <ng-template #enabled>
      <button (click)="showInfoWindow(application)" class="btn btn-sm btn-primary">
        info
      </button>
    </ng-template>
  </p>
</td>
Working example: Stackblitz
Update: debug current index
It's much easier and quicker to directly render the index in the template rather than to console.log the current index.
<td *ngFor="let application of applications; let index=index" style="padding: 5px; border-right: 1px solid #000000;">
  Current index: {{ index }}
  ...
</td>
If you however insist on printing into console, you could define a function in the controller and pass the current index to it using interpolation.
Controller
printIndex(index: number) {
  console.log(index);
}
Template
<td *ngFor="let application of applications; let index=index">
  {{ printIndex(index) }}
  ...
</td>
But beware: this might extremely misleading as the function will be triggered for each change detection cycle with default change detection strategy.